m 


LIBRARY    OF 

HENRY  C.  FALI — 


and  KATHARINE   A.  FALL 

Plumber  l^  8 

D^<?  0/  Purchase    ^A^nS.    £0    #3 

C05^  N 


■   .'V^;v.  ■ 


a 


■ 


DESCRIPTIVE  GEOMETRY, 


AS    APPLIED   TO 


THE  DRAWING  OF 


FORTIFICATION    AND    STEREOTOMY. 


FOR    THE    USE   OJ" 


THE  CADETS  OF  TEE  U.  S.  MILITARY  ACADEMY. 


BT 

D.  H.  MAHAN,  LL.L\, 

PE0FE8S0R   OF  FOETTFIOATION,  CIVIL  BNGIKEEBING,  *0.,  VNITED  STATES   illLITlET   AOAI>*m 


NEW   EDITION,   "WITH   ADDITIONS. 


NEW  YORK: 

JOHN   WILEY  &   SONS, 

15  ASTOR  PLACE. 

1883. 


■Entered,  according  to  Act  of  Congress,  in  the  yeai  1884,  by 

D.  H.  MAHAN, 

la  the  Clerk'*  Office  of  the  District  Court  of  the  United  States  for  the  Southern  DUtrloJ 

of  New  York. 


Trow's 
Printing  and  Hookrinding  Co., 

PRJN  ll-.KS     A  ,  1'liRS, 

205-213  East  12th  St., 

NEW    YUKK. 


PREFACE. 

Tile  subjects  of  the  following  pages  have  been 
taught  orally  at  the  Militaiy  Academy  for  many  years ; 
but,  for  the  saving  of  time,  and  the  convenience  of 
the  pupils,  it  has  been  thought  best  to  clothe  them 
in  a  printed  dress ;  and  as,  in  this  form,  the  volume 
might  be  found  useful  in  other  schools,  as  an  appli- 
cation of  descriptive  geometry  to  practical  questions, 
it  was  also  thought  well  to  have  it  published. 


ONE  PLANE  DESCRIPTIVE  GEOMETRY 


AS   APPLIED   TO 


FORTIFICATION  DRAWING. 


1.  The  method  now  in  general  use,  among  military  en- 
gineers, for  delineating  the  plans  of  permanent  fortifications, 
is  similar  to  the  one  which  had  been  previously  employed 
for  representing  the  natural  surface  of  ground  in  topograph- 
ical and  hydrographical  maps ;  and  which  consists  in  projec- 
ting, on  a  horizontal  plane  at  any  assumed  level,  the  bounding 
lines  of  the  surfaces  and  also  the  horizontal  lines  cut  from 
them  by  equidistant  horizontal  planes,  the  distances  of  these 
lines  from  the  assumed  plane  being  expressed  numerically 
in  terms  of  some  linear  measure,  as  a  yard,  a  foot,  &c. 

2.  Plane  of  Reference  or  Comparison.  The  assum- 
ed horizontal  plane  upon  which  the  lines  are  projected  is 
termed  the  plane  of  comparison  or  plane  of  reference,  as  it 
is  the  one  to  which  the  distances  of  all  the  lines  from  it  are 
referred,  and  as  it  serves  to  compare  these  distances  with 
each  other  and  also  to  determine  the  relative  positions  of 
the  lines. 

3.  References.  The  numbers  which  express. the  dis 
tances  of  points  and  lines  from  the  plane  of  comparison  are 
termed  references.  The  unit  in  which  these  distances  are 
expressed  is  usually  the  linear  foot  and  its  decimal  divisions. 

As  the  position  assumed  for  the  plane  of  comparison  is 
arbitrary,  it  may  be  taken  either  above  or  below  every  point 
of  the  surfaces  to  be  projected.  In  the  French  military  ser- 
vice it  is  usually  taken  above,  in  our  own  below  the  surfaces. 
The  latter  seems  the  more  natural  and  is  also  more  conveni- 
ent, as  vertical  distances  are  more  habitually  estimated  from 
below  upwards  than  in  the  contrary  direction.  Each  of 
these  methods  has  the  advantage  of  requiring  but  one  kind 
of  symbol  to  be  used,  viz :  the  numerals  expressing  the  ref 


2  ONE   PLANT-    DESCRIPTIVE   GEOMETBT. 

erenccs;  whereas,  if  the  plane  of  comparison  were  so  taken 
that  some  of  the  points  or  lines  projected  should  lie  on  one 

of  it  and  some  on  the  other,  it  would  be  then  necessary 
i  i  use,  in  connection  with  the  references,  the  algebraic  sym- 
bols plus  or  minus  to  designate  the  points  above  the  plane 
from  those  below  it. 

As  the  distances  of  all  points  are  estimated  from  the 
plane  of  comparison,  the  reference  of  any  point  or  line  of 
this  plane  will  therefore  be  zero,  (0.0);  that  of  any  point 
above  it  is  usually  expressed  in  feel  :  decimal  parts  of  a  foot 
being  used  whenever  the  reference  is  not  an  entire  number. 
When  the  reference  is  a  whole  number  it  is  written  with  one 
decimal  place,  thus  (25.0);  and  when  a  brok<  n  number  with 
at  least  two  decimal  places,  thus  (3.70),  (15.63).  In  writing 
the  reference  the  mark  used  to  designate  the  linear  unit  is 
omitted,  in  order  that  the  numbers  expressing  references 
may  not  be  mistaken  for  those  which  may  he  put  upon  the 
drawing  to  express  the  horizontal  distances  between  points. 

The  references  of  horizontal  lines  are  written  along  and 
upon  the  projections  of  these  lines.  All  other  references 
are  written  as  nearly  as  practicable  parallel  to  the  bottom  bor- 
der of  the  drawing,  for  the  convenience  of  reading  them 
without  having  to  shift  the  position  of  the  sheet  on  which 
the  drawing  is  made. 

This  method  of  representing  the  projections  of  objects 
on  one  plane  alone  has  given  rise  to  a  very  useful  modifica- 
tion of  the  one  of  orthogonal  projections  on  two  planes,  and 
has  been  denominated  one  mane  descriptive  geometry}  the 
plane  of  comparison  being  the  sole  plane  of  projection ;  and 
the  references  taking  the  place  of  the  usual  projections  on  a 
vertical  plane.  By  this  modification  the  number  of  lines  to 
be  drawn  is  less;  the  graphical  constructions  simplified; 
and  the  relations  of  the  parts  is  more  readily  seized  upon, 
as  the  eye  is  confined  to  the  examination  of  one  set  of  pro- 
jections alone.  • 

But  the  chief  advantage  of  it  consists  in  its  application 
to  the  delineation  of  objects,  like  works  of  permanent  forti- 
fication, where,  from  the  great  disparity  of  the  horizontal 
extent  covered  and  the  vertical  dimensions  of  the  parts,  a 
drawing,  made  to  a  scale  which  would  give  the  horiz< 
distances  with  accuracy,  could  not  in  most  cases  render  the 
vertical  dimensions  with  any  approach  to  the  same  de 
of  accuracy;  or,  if  made  to  a  scale  which  would  admit  of 
the  vertical  dimensions  being  accurately  determined,  would 
require  an  area  of  drawing  surface,  to  render  the  horizontal 
dimensions  to  the  same  scale,  which  would  exceed  the  con- 


ONE   PLANE   DESCRIPTIVE   GEOMETRY.  6 

venient  limits  of  practice.  Taking  for  example  an  ordinary 
scale  used  for  drawing  the  plans  of  permanent  fortifications 
of  one  inch  to  fifty  feet,  or  the  scale  g-^,  the  details  of  all 
the  bounding  surfaces  can  be  determined  with  accuracy  to 
within  the  fractional  part  of  a  foot,  whereas  a  vertical  pro- 
jection to  the  same  scale  would  be  altogether  too  small  for 
the  same  purposes. 

4.  Point  and  Right  Line.  To  designate  the  position 
of  a  point,  PI.  1,  Pig.  1,  the  projection  of  the  point-and  its 
reference  are  enclosed  within  a  bracket,  thus  (28.50).  This 
expresses  that  the  vertical  distance  of  the  point  from  the 
plane  of  reference  is  28  feet  and  fifty-hundredths  of  a  foot. 
The  position  of  a  right  line  oblique  to  the  plane  of  reference 
is  designated  by  the  projection  of  the  line,  and  the  references 
of  any  two  of  its  points.  Thus  in  Fig.  1  the  points  a  and 
b,  upon  the  projection  of  the  right  line,  with  their  respective 
references  (25.15)  and  (28.50),  determine  the  position  of  the 
line  with  respect  to  the  plane  of  reference. 

"When  the  line  is  horizontal,  or  parallel  to  the  plane  of 
reference,  its  projection,  with  the  reference  of  one  of  its 
points,  will  be  sufficient  to  designate  it,  and  fix  its  position 
with  respect  to  the  plane  of  reference.  Thus  in  Fig.  1  the 
reference  (25.15),  written  upon  the  projection  of  the  line, 
expresses  that  the  line  is  horizontal,  and  25.15  feet  from  the 
plane  of  reference. 

5.  For  the  convenience  of  numerical  calculation,  the  po- 
sition of  a  line,  with  respect  to  the  plane  of  reference,  is 
often  expressed  in  terms  of  the  natural  tangent  of  the  angle 
it  makes  with  this  plane;  but  as  this  angle  is  the  same  as 
that  between  the  line  and  its  projection,  its  natural  tangent 
can  be  expressed  by  the  difference  of  level  between  any  two 
points  of  the  line,  divided  by  the  horizontal  distance  between 
the  points.  Now,  as  the  difference  of  level  between  any 
two  points  of  the  line  is  the  same  as  the  difference  of  the 
references  of  the  points,  and  the  horizontal  distance  between 
them  is  the  same  as  the  horizontal  projection  of  the  portion 
of  the  line  between  the  same  points,  it  follows,  that  the  nat- 
ural tangent  of  the  angle  which  the  line  makes  with  the 
plane  of  reference  is  found  by  dmiding  the  difference  of  the 
references  of  the  points  by  the  distance  in  horizontal  projec- 
iton  between  them. 

The  vulgar  fraction  which  expresses  this  tangent  is  term- 
ed the  inclination,  or  declivity  of  the  line.  Thus  the  frac- 
tion i  would  express  that  the  horizontal  distance  between 
any  two  points  is  six  times  the  vertical  distance,  or  difference 
of  their  references ;  the  fraction  §,  that  the  vertical  distaiice 


4  ONE   PLANE   DESCRIPTIVE   GEOMETRY. 

bttween  any  two  points  is  two-thirds  the  horizontal  distance ; 
the  denominator  of  the  fraction,  in  all  cases,  representing 
the  number  of  parts  in  horizontal  projection,  and  the  nume- 
rator the  corresponding  number  of  parts  in  vein Meal  distance, 

When  the  position  of  a  line  is  designated  in  this  way,  it  ie 
said  to  be  a  line  whose  inclination  or  declivity  is  one-sixth, 
two-thirds,  ten  on  one,  &c,  or  simply,  a  line  of  one-s'uth,  &c. 

6.  Having  the  declivity  of  a  line,  the  difference  of  refer- 
ence of  any  two  of  its  points,  the  projections  of  which  are 
given,  will  be  found  by  multiplying  the  horizontal  distance 
between  them  by  the  fraction  which  expresses  this  declivity ; 
in  like  manner  the  horizontal  distance  of  any  two  points 
will  be  obtained  by  dividing  the  difference  of  their  references 
by  this  fraction. 

To  obtain  therefore  the  reference  of  a  point  of  a  line, 
having  its  projection,  the  horizontal  distance  between  it  and 
that  of  some  other  known  point  of  the  line  must  be  deter- 
mined from  the  scale  of  the  drawing  by  which  the  horizontal 
distances  are  measured ;  this  distance  expressed  in  numbers, 
being  multiplied  by  the  fraction  which  expresses  the  declivity 
of  the  line,  will  give  the  difference  of  reference  of  the  two 
points ;  the  required  reference  of  the  point  will  be  found  by 
subtracting  this  product  from  the  reference  of  the  known 

f)oint,  if  it  is  higher  than  the  one  sought,  or  adding  if  it  ie 
ower.  Thus  let  (25.15)  be  the  reference  of  a  known  point 
higher  than  the  one  sought;  the  horizontal  distance  between 
the  points  being  35.75  feet,  and  the  inclination  of  the  line 
TV ;  then  35.75  x  T\  =  3.575  will  be  the  difference  of  refer- 
ence of  the  points,  and  25.15  —  3.575  =  21.575,  the  required 
reference.  The  converse  of  this  shows  that  the  horizontal 
distance  between  two  points  on  this  line  whose  difference  of 
reference  is  3.575  will  be  3.575 -hTV= 35.75  feet. 

7.  When  the  projection  of  a  line  is  divided  into  equal 
parts,  each  of  which  corresponds  to'  a  unit  in  vertical  dis- 
tance, and  the  references  of  the  points  of  division  are  written, 
it  is  termed  the  scale  of  declivity  of  the  line.  In  constructing 
the  scale  of  declivity  of  a  line,  the  entire  references  are  alone 
put  down  ;  one  of  the  divisions  of  the  equal  parts  being  sub- 
divided into  tenths,  or  hundredths  if  necessary,  so  as  to  give 
the  fractional  parts  of  the  references  corresponding  to  an\ 
fractional  part  of  an  entire  division. 

8.  The  true  length  of  any  portion  of  an  oblique  line  be- 
tween two  given  points  is  evidently  the  hypothenuse  of  a 
right  angle  triangle  of  which  the  other  two  sides  are  the  dif 
ference  of  reference  ot  the  points,  and  their  horizontal  dis- 
tance. 


ONE   PLANE   DESCRIPTIVE   GEOMETRY.  5 

9.  Plane.  The  position  of  a  plane  oblique  to  the  plane 
of  reference  may  be  determined  either  by  the  projections  and 
references  of  three  of  its  points;  by  the  projections  and  de- 
clivity of  two  lines  in  it  oblique  to  the  plane  of  reference ; 
or  by  the  projection  of  two  or  more  horizontal  lines  of  the 
plane  with  their  references. 

The  more  usual  method  of  representing  a  plane  is  by  the 
projection^  on  the  plane  of  reference  of  the  horizontal  lines 
determined  by  intersecting  it  by  equidistant  horizontal  planes. 
These  projections  are  termed  horizontals  of  the  plane,  those 
usually  being  taken  the  references  of  which  are  entire  numbers. 

10.  If  in  a  given  plane  a  line  be  drawn  perpendicular  to 
any  horizontal  line  in  it,  the  projection  of  this  line  on  the 
plane  of  reference  will  be  also  perpendicular  to  the  projec- 
tions of  the  horizontals.  The  angle  of  this  line  with  the 
plane  of  reference  is  evidently  the  same  as  that  of  the  given 
plane  with  it,  and  is  greater  than  the  angle  between  any 
other  line  drawn  in  the  plane  and  the  plane  of  reference. 
This  line  is,  on  this  account,  termed  the  line  of  greatest  de- 
clivity of  the  plane. 

11.  If  the  scale  of  declivity  of  the  line  of  greatest  de- 
clivity be  constructed,  it  will  alone  serve  to  fix  the  position 
of  the  plane  to  which  it  belongs,  and  to  determine  the  refer- 
ence of  any  point  of  the  plane  of  which  the  projection  is 
given.  For  since  the  horizontals  are  perpendicular  to  the 
scale  of  declivity,  the  point  where  the  horizontal  drawn 
through  the  given  projection  of  a  point  in  the  plane  cuts 
this  line  will  determine  upon  the  scale  the  reference  of  the 
horizontal,  and  therefore  that  of  the  point. 

12.  The  inclination  or  declivity  of  a  plane  with  the  plane 
of  reference  may  be  expressed  in  the  same  way  as  the  incli- 
nation of  its  line  of  greatest  declivity.  Thus  a  plane  of  one- 
fourth;  a  plane  of  twenty  on  one;  a  plane  of  two-thirds, 
express  that  the  natural  tangents  of  the  angle  between  the 
planes  and  the  plane  of  reference  are  respectively  represent- 
ed by  the  fractions  i,  -2T%  and  f. 

13.  The  horizontal  distance  between  any  two  horizontal 
lines  in  a  plane,  the  angle  of  which  is  given,  can  be  found 
in  the  same  way  as  the  horizontal  distance  between  two 
points  of  a  line,  the  inclination  of  which  is  given,  Art.  7, 
by  dividing  the  difference  of  the  reference  of  the  two  hori- 
zontal lines  by  th$  fraction  representing  the  declivity  of  the 
plane ;  in  like  manner  the  difference  of  references  of  any 
two  horizontal  lines  will  be  obtained  by  multiplying  their 
horizontal  distance  by  the  same  fraction. 

14.  To  distinguish  the  scale  of  declivity,  P\.  1,  Fvg.  2, 


6  ONE   PLANE   DESCRIPTIVE    GEOMETRY. 

from  any  other  line  of  a  plane,  it  is  always  represented  by 
two  fine  parallel  lines,  drawn  near  each  other,  and  crossed 
at  the  points  of  division,  where  the  references  are  written, 
by  short  liues  which  are  portions  of  the  corresponding  hori- 
zontals. 

With  the  foregoing  elements  the  usual  problems  of  the 
right  line  and  plane  can  be  readily  solved. 

15.  Problems  of  the  Right  Line  and  Plaile. 
Prob.  1,  PI.  1,  Fig.  3.    Ha/ovng  the  prqjt  ct/ions  and  refer 

end  8  of  two  lines  that  intersect,  tofnd  tin  a/ngh  between  t)<>  m. 

Let  at)  be  the  projection  of  one  of  the  lines,  the  refer- 
ences of  two  of  its  points  (10.30)  and  (4.90)  being  given 
cd  the  projection  of  the  other  line,  (10.30),  and  (5.0)  being 
the  references  of  two  of  its  points;  (10.30)  being  the  point 
of  intersection  of  the  two  lines. 

Find  on  each  of  the  lines,  Art.  7,  a  point  having  the 
name  reference  (7.0).  The  line  joining  these  tAvo  points 
will  be  horizontal,  and  projected  into  its  true  length  ;  taking 
this  line  as  the  base  of  a  triangle  of  which  the  other  two  sides 
are  respectively  the  true  lengths  of  the  portions  of  the  two 
given  lines  projected  between  (10.30)  and  (T.0),  Art.  7,  the 
angle  at  the  vertex  will  be  the  one  required. 

16.  Prob.  2,  Pig.  4.  Through  a  point  to  draw  a  Una 
parallel  to  «  given  line. 

Let  c  (7.50)  be  the  projection  of  the  point ;  ab  that  of 
the  given  line  of  which  the  two  points  (7.0)  and  (9.0)  are 
known. 

Through  o  drawing  cd  parallel  to  ab,  this  will  be  the 
projection  of  the  required  line ;  and  as  its  declivity  is  the 
same  as  that  of  the  given  line,  it  will  be  only  necessary  to 
set  off  from  c  towards  d,  the  same  distance  as  between  (7.0) 
and  (9.0),  to  obtain  a  point  (9.50)  as  far  above  (7.50)  as  (9.0) 
is  above  (7.0). 

17.  Prob.  3,  Fig.  5.  Through  a  point  in  a  plane  to  draw 
a  line  in  the  plane  with  a  given  inclination. 

Let  cd  be  the  scale  of  declivity  of  the  given  plane,  and 
a  (5.50)  the  given  point ;  and  suppose,  for  example,  that  the 
declivity  of  the  plane  is  \  and  that  of  the  required  line  is  TV. 

Draw  the  horizontal  of  the  plane  (5.50)  which  ] 
through  the  point,  and  any  other  horizontal,  as  (7.0).  The 
projection  of  the  required  line  will  pass  through  a,  and  the 
portion  of  it  between  the  two  horizontals  will  be  equal,  Art. 
f>,  to  the  difference  of  their  references,  or  1.5  ft.  divided  by 
the  fraction  which  represents  the  inclination  of  the  required 
line.  Describing,  therefore,  from  a,  an  arc,  with  this  dis- 
tance ac  or  1.5  -i-  T\  =  15  ft.  as  a  radius,  and  joining  the 


ONE   PLANE    DESCRIPTIVE-  GEOMETRY.  7 

point  b,  where  it  cuts  the  horizontal  (7.0),  with  #,  this  will 
be  the  projection  of  the  required  line. 

18.  Prob.  4,  PI.  1,  Fig.  6.  Saving  three  points  of  a 
plane,  to  construct  its  horizontals  and  scale  of  declivity. 

Let  a  (12.0),  b  (15.25),  and  c  (15.50),  be  the  projections  of 
the  three  points.  Join  a  with  the  other  two,  and  construct 
the  scales  of  declivity  of  the  lines  of  junction,  Art.  6.  The 
lines  joining  the  same  references  on  these  two  scales  will  be 
horizontals  of  the  required  plane.  Its  scale  of  declivity  is 
constructed  by  drawing  two  parallel  lines  perpendicular  to 
the  horizontals,  and  writing  the  references  of  the  points 
where  they  intersect  the  horizontals. 

19.  Prob.  5,  PI.  1,  Pig.  1.  To  find  the  horizontals  of 
a  plane  passed  through  a  given  line  and  parallel  to  another 
line. 

Let  ab  and  cdhe  the  projections  of  the  two  lines.  From 
a  point  (10.0)  on  cd  draw  a  line  df  Prob.  2,  parallel  to  ab; 
and  by  Prob.  4  find  the  horizontals  of  the  plane  of  df  and 
cd ;  these  will  be  the  required  horizontals. 

20.  Prob.  6,  PI.  1,  Fig.  8.  To  find  the  horizontals  of  a 
plans  the  declivity  of  which  is  given,  and  which  passes 
through  a  given  line. 

Let  bd  be  the  scale  of  declivity  of  the  given  line,  and 
suppose,  for  example,  the  declivity  of  the  line  to  be  T'j  and 
that  of  the  required  plane  to  be  }. 

Since  the  horizontals  of  the  plane  must  pass  through  the 
points  of  the  line  having  the  like  references,  and  as  the  dis- 
tance in  projection  between  any  two  of  them,  Art.  13,  will 
be  equal  to  the  difference  of  their  references  divided  by  the 
fraction  giving  the  declivity  of  the  plane,  it  follows  that  to 
find  the  one  drawn  through  b  (14.0),  for  example,  it  will  be 
simply  necessary  to  describe  from  any  other  point,  as  a 
(12.0),  an  arc  of  a  circle,  with  a  radius  of  12  ft.,  equal  to 
the  quotient  just  mentioned,  and  to  draw  a  tangent  to  this 
arc  from  b.  If  any  other  horizontal,  as  (16.0),  is  required, 
which  would  not  intersect  the  projection  of  the  given  line 
within  the  limits  of  the  drawing;  any  two  points,  as  (12.0) 
and  (14.0),  for  example,  may  be  taken  as  centres,  and  two 
arcs  be  described  from  them,  with  radii  of  12  and  24  ft., 
calculated  as  above,  and  a  line  be  drawn  tangent  to  the 
arc;  this  tangent  will  be  the  required  horizontal. 

21.  Prob.  7,  PL  1,  Fig.  9.  Hiving  either  the  horizontals 
or  the  scales  of  declivity  of  two  planes,  to  find  their  intersec- 
tion. 

'       Join  the  points  ah  where  any  two  horizontals,  as  (12.0) 
and  (14.0),  in  one  plane  intersect  the  corresponding  horizon- 


8  ONE   PLANE    DESCRIPTIVE    GEOMETRY. 

tals  of  the  other,  and  the  line  so  drawn  will  be  the  projection 
of  the  required  intersection. 

22.  When  the  horizontals  of  the  two  planes  are  parallel, 
or  when  they  are  so  nearly  parallel  that  their  points  of  in- 
tersection cannot  be  accurately  found,  the  following  method 
may  be  taken :  Draw  any  two  parallel  lines  as  cd,  cd',  PI. 
1,  Fig.  10 ;  these  may  be  considered  as  the  horizontals  of  an 
arbitrary  plane,  and  having  the  same  references,  (12.0)  and 
(14.0),  as  the  two  corresponding  horizontals  in  each  of  the 
given  planes.  The  intersections  of  the  horizontals  of  the 
arbitrary  plane  with  those  of  the  given  planes  will  determine 
two  lines,  mn}  m'n' ,  which,  being  the  projections  of  the  in- 
tersections of  the  given  planes  with  the  arbitrary  plane, 
will,  by  their  intersection  o,  determine  the  projection  of  a 
point  common  to  the  three  planes,  and  therefore  a  point  of 
the  projection  of  the  intersection  of  the  two  given  planes. 
Assuming  any  other  two  parallels  ab,  a 'I',  as  the  horizontals 
of  another  arbitrary  plane ;  finding  in  like  manner  the  point 
o  and  joining  o  and  o'  by  a  line,  this  will  be  the  required 
projection. 

When  the  horizontals  of  the  two  planes  are  parallel,  one 
point,  as  o,  will  be  sufficient  to  determine  the  required  pro- 
jection, as  it  will  be  parallel  to  the  horizontals. 

23.  Prob.  8,  PI.  1,  Fig.  11.  To  find  where  a  given  line 
pierces  a  given  plane. 

Through  the  projections  of  any  two  points  of  the  given 
line,  as  m/,  n' ,  having  the  same  references,  (12.0),  (14.0),  as 
two  horizontals  of  the  given  plane,  draw  two  parallel  lines, 
ah,  a'b ',  which  may  be  taken  as  the  horizontals  of  an  arbitrary 
plane.  The  projection  of  the  line  of  intersection,  m,  of 
this  plane  with  the  given  plane  being  determined  by  Prob. 
7,  the  point  o  where  it  intersects  the  projection  of  the  line 
m'n'  will  be  the  projection  of  the  required  point,  the  refer- 
ence of  which  can  be  found  from  the  scale  of  the  plane. 

24.  Prob.  9,  PI.  1,  Fig.  12.  To  draw  from  a  given 
point  a  perpendicidar  to  a  given  plane,  and  find  its  length. 

Let  a  (12.0)  be  the  projection  of  the  given  point ;  and 
let  the  given  plane  be  represented  by  its  scale  of  declivity. 

The  projection  of  the  required  perpendicular  will  pass 
through  a,  and  be  parallel  to  the  scale  of  declivity  of  the 
given  plane.  The  angle  which  it  makes  with  the  plane  of 
reference  is  the  complement  of  that  between  this  plane  and 
the  given  plane ;  its  tangent  therefore  will  be  the  reciprocal 
of  the  tangent  of  that  of  the  given  plane. 

Drawing  therefore  through  a  the  line  ac  parallel  to  bd, 
and  constructing  its  scale  of  declivity,  Art.  7,  this  will  be 


ONE    PLANE    DESCRIPTIVE    GEOMETRY.  9 

the  projection  of  the  required  perpendicular.  The  projec- 
tion of  the  point  o  where  it  pierces  the  given  plane  is  found 
by  Prdb.  8,  and  the  true  length  of  the  perpendicular  by 
Art.  8. 

25.  Geometrical  and  Irregular  Surfaces. 

All  other  surfaces  may,  like  the  plane,  Art.  7,  be  repre- 
sented by  the  projections  on  the  plane  of  reference  of  the 
curves  or  lines  cut  from  them  by  equidistant  horizontal 
planes,  together  with  the  references  of  these  curves ;  as  many 
of  these  projections  being  drawn  as  may  be  requisite  to  de- 
termine all  the  points  of  the  surface  with  accuracy;  and 
their  references  being  written  in  the  same  way  as  those  of 
the  horizontals  of  a  plane. 

In  the  more  simple  geometrical  surfaces,  a  single  hori- 
zontal curve,  with  the  projection  of  some  point  or  line  of 
the  surface,  will  alone  suffice.  For  example,  the  cone  may  be 
represented  by  the  projection  and  reference  of  any  curve  cut 
from  it  by  a  horizontal  plane,  with  the  projection  and  refei 
ence  of  its  vertex ;  a  cylinder  by  the  projection  and  reference 
of  a  like  curve,  with  the  projection  and  reference  of  its  axis, 
or  of  one  of  its  right  line  elements ;  a  sphere  by  the  projec- 
tion and  reference  of  its  centre  and  that  of  its  great  circle 
parallel  to  the  plane  of  reference. 

26.  This  method  of  projection  is  more  particularly  ad- 
vantageous in  the  representation  of  irregular  surfaces  which, 
like  the  natural  surfaces  of  ground,  for  example,  are  not  sub- 
mitted to  any  geometrical  law,  and  in  solving  the  various 
problems  of  tangent  and  secant  planes  to  surfaces  of  this 
character.  These  surfaces  can,  for  the  most  part,  be  alone 
represented  by  the  projections  of  the  horizontal  curves  cut 
from  them  by  equidistant  horizontal  planes,  and  by  suppos- 
ing the  zone  of  the  real  surface  contained  between  any  two 
horizontal  curves  to  be  replaced  by  an  artificial  zone,  sub- 
jected to  some  geometrical  law  of  generation,  which  shall 
give  an  approximation  to  the  real  surface  sufficiently  accu- 
rate  for  the  object  in  view.  The  usual  method  of  doing  this 
is  to  take  two  consecutive  horizontal  curves  as  the  directrices 
of  the  artificial  surface  of  the  zone,  and  to  move  a  right  line 
bo  as  to  continually  intersect  each  of  them,^  and  be  perpen- 
dicular to  the  consecutive  tangents  to  one  of  them,  the  upper 
one  being  usually  taken  for  this  last  condition. 

If  in  PI.  1,  Fig.  13,  for  example,  (6.0),  (7.0),  &c,  are  the 
orojections  of  the  horizontals  of  a  surface,  the  zone  between 
the  curves  (6.0)  and  (7.0)  may  be  replaced  by  an  artificial 
surface,  the  position  of  the  projection  of  the  generatrix  of 
which,  at  any  point  of  the  upper  curve  (7.0),  will  be  deter- 


10  ONE   PLANE   DESCRIPTIVE    ^""JiOMETBY. 

mined  by  constructing  the  horizontal  tangent  at  that  point 
as  0,  for  example,  and  drawingoS  perpendicular  to  it  and 

intersecting  the  lower  curve.  The  position  of  the  generatrix 
a'b'  at  any  other  point  a'  is  constructed  in  like  manner. 

'27.  To  obtain  any  horizontal  of  the  artificial  zone  inter- 
mediate to  the  two  directrices,  it  will  be  only  necessary  to 
construct  several  positions  of  the  generatrix,  and  to  find  on 
these  the  points  having  the  same  reference  as  the  required 
curve.  The  horizontal  of  the  surface  (6.50),  for  example, 
will  bisect  the  projections  of  the  generatrix  in  its  varioua 
positions. 

Problems  of  Irregular  Surfaces  and  the  Right 
Line  and  Plane. 

28.  Proh.  10,  PI.  1,  Fig.  14.  Through  a  given  point  in 
a  vertical  plam  vahich  intersects  a  surface,  to  draw  a  tangent 
to  the  curve  of  intersection  of  the  plane  and  surface. 

Let  a  (5.50)  be  the  given  point,  and  ab  the  trace  on  the 
plane  of  reference  of  the  given  plane.  The  points  where 
this  trace  intersects  the  horizontal  curves  of  the  surface  will 
be  the  projections  of  points  of  the  curve  cut  from  the  surface 
by  the  plane. 

Let  any  arbitrary  line  as  ac  be  now  drawn  through  a, 
and  its  scale  of  declivity  be  constructed ;  and  let  lines  be 
drawn  between  the  points  having  the  same  references  on  ac 
and  on  the  horizontal  curves  where  ah  intersects  them.  These 
lines  will  be  the  projections  of  horizontal  lines  and  will  gen- 
erally make  different  angles  with  ac.  The  one  as  (T.'M, 
which  makes  the  smallest  angle  with  it,  towards  the  descend- 
ing portion,  will  determine  the  projection  o  of  the  tangential 
point.  For,  construct  the  scale  of  declivity  of  the  line  of 
which  a  (5.50)  is  the  projection  of  one  point,  and  o  (7.0),  on 
ab,  another.  Comparing  now  the  references  of  the  points 
on  the  line,  and  which  is  assumed  as  the  projection  of  the 
required  tangent,  with  the  references  of  the  points  of  the 
curve  having  the  same  projection,  it  will  at  once  be  evident 
that  these  two  lines  have  only  the  point  projected  in  (7.0)  in 
common,  and  that  every  other  point  of  the  right  line,  of 
which  adb  is  the  projection,  is  exterior  to  the  curve,  and 
therefore  the  line  itself  must  be  tangent  to  the  curve  at  the 
point  determined  as  above. 

29.  Proh.  11,  PI.  1,  Fig.  15.  To  construct  the  elements 
of  a  cone,  with  a  given  vertex,  which  shall  envelope  a  given 
surface. 

Let  (10.0),  &c,  be  the  horizontals  of  the  given  surface ; 
and  a  (6.0)  the  projection  of  the  vertex  of  the  cone. 

From  a,  draw  lines  ab,  ab' ,  &c,  as  the  horizontal  traces 


0XE   PLANE    DESCRIPTIVE    GEOMETRY.  11 

of  vertical  planes  whicli  pass  through  the  vertex  and  inter 
sect  the  surface.  Construct,  by  Prob.  10,  the  tangents  from 
a  to  the  curves  cut  from  the  surface  by  the  planes  ab,  &c. 
These  tangents  will  be  the  required  elements. 

30.  Prob.  12,  PI  1,  Fig.  15.  To  find  the  curve  of  in- 
tersection of  a  cone  enveloping  a  given  surface  by  a  Jwrizon- 
tal  plane. 

Let  (9.0)  be  the  reference  of  the  given  horizontal  plane. 
Having  found,  by  Probs.  11  and  12,  the  elements  of  the  cone, 
and  constructed  the  scale  of  declivity  of  each  one ;  then 
joining  the  points  o,  o',  o",  having  the  same  reference  on 
each  scale  as  the  given  horizontal  plane,  a  continuous  line 
mo"o'on  will  be  obtained,  which  will  be  the  projection  of 
the  points  where  the  elements  pierce  the  given  plane,  and 
therefore  the  projection  of  the  required  intersection. 

31.  Prob.  13,  PI.  2,  Fig.  1.  A  limited  extent  of  surface 
being  given,  and  a  point  exterior  to  it,  to  find  the  limits  with- 
in which  planes  may  be  passed  through  this  point  and  lie 
above  all  the  given  surface. 

Let  a  (8.0)  be  the  projection  of  the  given  point ;  (10.0), 
(9.0),  &c,  the  horizontals  of  the  given  surface,  the  limits  of 
which  are  the  sector  contained  within  the  arc  BDC,  and 
the  two  radii  aB  and  aC. 

Taking  a  as  the  vertex  of  a  cone  which  shall  envelope 
the  given  surface,  the  elements  of  this  cone  can  be  found  by 
Probs.  11  and  12.  Any  plane  tangent  to  this  cone,  which 
does  not  intersect  the  surface  within  the  given  limits,  will 
satisfy  the  conditions  of  the  problem. 

From  the  position  of  the  vertex  of  the  cone  with  respect 
to  the  surface,  it  will  be  seen  that  a  horizontal  plane,  passed 
through  the  vertex,  will  cut  from  the  cone  two  elements 
which  will  be  projected  in  the  two  horizontals  ab'  and  ah" 
(8.0)  of  the  cone,  the  first  of  which  will  be  tangent  to  the 
horizontal  (8.0)  of  the  surface,  and  the  second  ab"  will 
pierce  the  surface,  where  the  limiting  arc  BDC  cuts  the 
same  horizontal  (8.0) ;  and  that  all  the  elements  projected 
within  the  angles  Bab'  and  Cab"  will  lie  below  the  horizon- 
tal plane  (8.0).  Now,  if  the  elements  within  these  angles 
be  prolonged  beyond  the  vertex,  they  will  form  two  portions 
of  cones  having  the  same  elements  as  the  portions  below  the 
vertex,  and  it  is  evident  that  any  plane  passed  tangent  to 
either  lower  portion,  as  b'aB,  within  one  of  these  angles, 
will  leave  this  portion  below  it,  and  the  corresponding  por- 
tion, formed  by  the  prolonged  elements,  above  it ;  and,  in 
order  that  thisplane  shall  satisfy  the  conditions  of  the  prob- 
lem, it  must  also  leave  the  portions  of  the  cone  within  the 


12  ONE    PLAITS    DESCRIPTIVE   GEOMETRY. 

angles  b'ab" ,  and  b"aC,  also  below  it.  The  same  reasoning 
applies  to  planes  passed  tangent  to  the  portions  of  the  cone 
within  each  of  the  other  two  angles.  It  is  therefore  evident 
that  a  plane,  which  shall  satisfy  the  conditions  imposed, 
must  leave  all  that  portion  of  the  cone  which  lies  above  the 
horizontal  plane  (8.0)  through  the  vertex,  below  it,  and  all 
the  prolonged  portions,  corresponding  to  the  portions  below 
the  plane  (8.0),  above  it. 

To  find  any  such  plane,  let  the  cone  be  intersected  by  a 
horizontal  plane,  as  (9.0),  by  Prob.  12.  This  plane  will  cut, 
from  the  portion  of  the  cone  within  the  angle  b'ab",  a  curve 
of  which  non'  is  the  projection ;  the  two  extreme  points  of 
this  curve,  within  the  limits,  being  at  the  points  nn',  where 
the  horizontal  (9.0)  of  the  surface  cuts  the  limiting  arc;  it 
will  also  cut,  from  each  of  the  prolonged  portions,  a  curve, 
the  one  mr,  and  the  other  m'r'  j  the  extreme  poini  m  of  m? 
being  on  the  prolongation  of  the  extreme  element  aC 'j  that 
m!  of  the  other  on  the  extreme  element  aB,  on  the  other 
side,  prolonged.  Having  obtained  these  three  curves,  let 
tangent  lines,  ms,  m's ',  be  drawn,  from  the  points  m  and  m ', 
to  the  curve  non' .  A  plane  passed  through  either  of  these 
tangents  and  through  the  corresponding  element  of  the  cone 
as  or  as',  drawn  through  the  tangential  point,  will  be  a  tan- 
gent plane  to  the  cone ;  and  as  either  of  these  planes  will 
leave  the  curve  non!  on  one  side  of  it,  and  the  two  curves 
mr,  and  m'r',  on  the  other,  it  will  leave  all  the  portion  of 
the  cone  corresponding  to  the  first  curve  below  it,  and  the 
portions  corresponding  to  the  other  curves  above  it ;  and 
will  therefore  satisfy  the  required  conditions.  The  same  will 
hold  true  for  any  tangent  plane  to  the  cone  along  any  ele- 
ment drawn  between  the  points  s  and  «'/  since  the  tangent 
drawn  to  any  point  of  the  curve  non',  between  the  points  s 
and  s',  will  leave  this  curve  on  one  side  of  it,  and  the  other 
two,  mr  and  m'r',  on  the  other. 

32.  Prob.  14,  PI.  1,  Fig.  16.  Hi/rough  a  given  line  to 
pass  a  plane  tangent  to  a  surface. 

1st.  Let  ab  be  the  projection  of  the  given  line,  and  (10.0), 
(9.0),  &c,  the  horizontals  of  the  surface.  From  the  points 
on  the  line,  as  (10.0),  &c,  draw  lines  tangent  to  the  horizon- 
tals having  the  same  references;  the  tangent  which  makes 
with  the  projection  of  the  line  the  least  angle  towards  the 
descending  portion,  will,  with  the  line,  determine  the  requir- 
ed plane. 

For,  let  the  tangent  (10.0)  be  the  one  which  makes  with 
ab  the  least  angle;  from  the  other  points,  (9.0),  eve.,  of  ab, 
draw  lines  parallel  to  the  tangent  (10.0);  these  lines  will  lie 


ONE   PLANE   AND   DESCRIPTIVE   GEOMETRY.  13 

in  the  plane  that  contains  this  tangent  and  ah,  and  will  be 
horizontals  of  this  plane ;  they  also  lie  respectively  in  the 
planes  of  the  horizontals  (9.0),  (8.0),  &c,  of  the  surface,  but, 
since  they  fall  exterior  to  these  horizontals,  it  follows  that 
their  plane  also  lies  exterior  to  every  horizontal  curve  of  the 
surface,  except  at  the  curve  (10.0),  and  where  it  touches  the 
surface  at  the  point  of  contact  of  its  horizontal  (10.0)  with 
this  curve. 

2d.  When  the  line  ah,  PI.  1,  Fig.  17,  is  horizontal,  let 
tangents  be  drawn  to  the  horizontal  curves  and  parallel  to 
ah.  These  tangents  may  be  regarded  as  the  elements  of  a 
cylinder  which  envelops  the  surface,  the  tangent  plane  to 
which  will  be  tangent  to  the  surface.  To  find  the  element 
of  contact  of  the  plane  and  cylinder,  let  the  cylinder  and 
given  line  be  intersected  by  an  arbitrary  vertical  plane,  of 
which  od  is  the  trace.  From  the  point  o,  (6.5),  where  the 
line  pierces  this  plane,  let  a  tangent  line  be  drawn  to  the 
curve  cut  from  the  cylinder  by  the  plane,  by  Proh.  10.  The 
point  of  contact  will  determine  the  position  of  the  element 
of  the  cylinder  along  which  the  plane,  through  ah,  will  be 
tangent ;  since  the  tangent  to  the  curve  projected  in  od,  with 
the  line  ah,  will  determine  the  tangent  plane  to  the  cylinder. 

3d.  "WTien  the  line  ah,  PI.  1,  Fig.  18,  is  so  nearly  hori- 
zontal that  tangents  cannot  be  drawn  from  its  points,  within 
the  limits  of  the  drawing,  to  the  horizontal  curves.  Let  any 
point  of  the  line,  as  o,  (7.0),  be  taken  as  the  vertex  of  a  cone 
enveloping  the  surface ;  a  plane  passed  through  the  line  and 
tangent  to  the  cone  will  be  tangent  to  the  surface. 

Find,  by  Probs.  10  and  11,  the  projection  mm  of  the 
curve  cut  from  this  cone  by  the  horizontal  plane  (8.0) ;  from 
the  point  (8.0)  of  ah  draw  a  tangent  to  mm.  This  tangent, 
with  the  line  ah,  will  determine  the  required  plane. 

33.  Proh.  15,  PI.  1,  Fig.  19.  To  find  approximately 
the  point  where  a  given  right  line  pierces  a  surface. 

Let  (8.0),  (9.0),  &c,  be  the  horizontals  of  the  surface,  and 
df  the  scale  of  declivity  of  the  line.  Through  any  two 
points,  as  a  (9.0)  and  c  (8.0),  draw  two  parallel  lines,  as  am 
and  en,  which  may  be  taken  as  the  horizontals  of  an  arbi- 
trary plane  passed  through  the  given  line.  Joining  the 
points  m,  n  where  the  horizontals  of  the  arbitrary  plane  in- 
tersect the  corresponding  horizontals  of  the  surface,  this 
line  mn  will  be  the  approximate  intersection  of  the  plane 
with  the  zone  of  the  surface  between  the  horizontals  (8.0) 
and  (9.0),  and  the  point  o  where  mn  intersects  df  will  be 
the  approximate  point  required. 


14  OXE    PLANE    DESCRIPTIVE    GEOMETRY. 

34.  Prdb.  16,  PI.  1,  Fig.  20.  To  >w2  Me  intersection 
of  a  plane  and  surface. 

Let  (10.0),  (9.0),  tfcc.,  be  the  horizontals  of  the  surface, 
ef  the  scale  of  declivity  of  the  plane. 

Draw  the  horizontals  of  the  plane  having  the  same  refer- 
ences as  the  horizontals  of  the  surface,  the  points  of  inter- 
section of  the  corresponding  lines  will  be  the  projections  of 
points  of  the  required  intersection. 

When  it  is  desired  to  find  a  point  of  the  curve  of  inter- 
section intermediate  to  two  horizontal  curves ;  if  the  refer- 
ence of  the  required  point  is  fixed,  it  will  be  necessary  to 
construct,  Art.  27,  the  horizontal  of  the  surface,  and  the 
horizontal  of  the  plane  having  this  reference ;  their  intersec- 
tion will  give  the  projection  of  the  required  point.  If  the 
reference  of  the  required  point  is  not  fixed,  draw  any  gene- 
ratrix, as  ao  of  the  zone  on  which  the  required  point  is  to  be 
found,  and  by  Prob.  8,  Fig.  11,  find  the  projection  of  the 
point,  as  o,  where  ac  pierces  the  given  plane;  this  will  be 
the  required  point. 

35.  Application  of  Preceding  Problems. 

The  following  problems  will  aid  as  illustrations  of  the 
preceding  subject  in  its  application  to  the  determination  and 
delineation  of  lines  and  surfaces. 

36.  Prob.  1,  PI.  2,  Fig.  2.  The  plane  of  site  of  a  ivork, 
the  exterior  line  and  scale  of  declivity  of  its  terre-plein  being 
given;  to  construct  the  plane  of  the  rampart-slope  and  its 
foot;  also  a  ramp  of  a  given  inclination  along  the  rampart- 
slope  leading  from  the  plane  of  site  to  the  terre-plein. 

Let  a  (74.50)  and  b  (76.0)  be  the  references  of  two  points 
on  the  exterior  line  of  the  terre-plein,  and  mn  its  scale  of 
declivity;  let  the  rampart-slope  be  ■§,  the  declivity  of  the 
ramp  |,  its  width  4.30  yards ;  and  the  plane  of  site  be  hori- 
zontal and  at  the  ref.  (60.0). 

The  foot  of  the  rampart-slope  lying  in  the  plane  of  site 
will  be  horizontal,  and  will  be  determined,  Prdb.  6,  Fig.  8, 
by  finding  the  line  of  the  slope  at  the  ref.  (60.0). 

Having  the  two  bounding  lines  of  the  rampart-slope,  the 
inner  line  cd  of  the  ramp  is  constructed,  by  assuming  a 
point  <?,  on  the  foot  of  the  rampart-slope,  as  the  point  of 
departure,  and  determining  the  line  of  £  drawn  from  c  on 
the  rampart-slope  by  Prdb.  3,  Fig.  5.  Having  found  this 
line,  which  is  also  the  line  of  greatest  declivity  of  the  ramp, 
the  exterior  line  ef  of  the  ramp  is  drawn  parallel  to  it,  and 
at  a  distance  4.30  yds.,  equal  to  the  width  assumed  for  the 
ramp.  The  horizontals  of  the  ramp  will  bo  perpendicular 
to  these  two  lines.     The  foot  of  the  ramp,  ce,  will  be  a  hor- 


ONE   PLANE    DESCRIPTIVE    GEOMETRY.  19 

izontal  line,  drawn  through  the  point  of  departure.  The 
top  of  it,  df,  will  he  determined  by  Prof).  7,  Fig.  9,  by  find- 
ing the  intersection  of  the  ramp  and  the  terre-plein,  one 
point  of  which  will  be  the  point  d  (76.30),  the  intersection 
of  the  inner  line  of  the  ramp  and  the  interior  line  of  the 
terre-plein. 

The  ramp  is  terminated  on  the  exterior,  by  passing  a 
plane  through  its  exterior  line  ef  having  the  same  slope  as 
the  rampart-slope.  This  plane  will  intersect  the  plane  of 
site  in  a  line  parallel  to  the  foot  of  the  rampart-slope,  and 
the  terre-plein  in  one  parallel  to  the  exterior  line  of  the 
terre-plein. 

37.  Prob.  18,  PI.  2,  Fig.  3.  Having  given  the  lines  of 
the  parapet  of  a  work  and  the  scales  of  declivity  of  the  planes 
of  its  interior  crest  and  terre-plein,  to  determine  the  lines  and 
surfaces  of  a  ba/rbette  in  its  salient  for  five  guns. 

Let  ab  be  the  scale  of  declivity  of  the  plane  of  the  interior 
crest,  which,  as  the  terre-plein  is  parallel  to  the  plane  of 
the  interior  crest  and  8  feet  below  it  estimated  vertically, 
will  also  serve  as  the  scale  of  declivity  of  the  terre-plein,  by 
subtracting  8  feet  from  the  references  of  the  former  to  obtain 
the  corresponding  references  of  the  latter.  Having  con- 
structed a  pancoupe  of  4  yds.  in  the  salient,  find  the  intersec- 
tion of  the  top  surface  of  the  barbette,  which  is  horizontal 
and  assumed  on  the  drawing  at  the  reference  (82.75)  with 
the  planes  of  the  interior  slope,  this  intersection  will  deter- 
mine the  foot  of  the  genouillere  of  the  barbette.  From  this 
last  line  at  the  pancoupe  set  back  along  the  capital  a  distance 
of  8  yds.,  and  from  the  extremity  of  this  line  draw  a  per- 
pendicular to  the  interior  crest  of  each  face.  The  pentag- 
onal figure  thus  marked  out  will  be  the  space  for  the  gun  in 
the  salient.  From  the  foot  of  each  of  the  perpendiculars 
set  off  along  the  faces  distances  of  12  yds.  for  the  lengths 
along  the  interior  crests  to  be  occupied  by  two  guns  on  each 
side  of  the  salient.  Setting  back  from  the  extremities  of 
these  two  last  distances  perpendiculars  to  the  interior 
crest  of  8  yds.  and  drawing  lines  through  the  extremities  of 
these  perpendiculars  parallel  to  the  interior  crests,  they  with 
the  two  perpendiculars  will  mark  out  the  exterior  bounding 
lines  of  the  barbette.  By  passing  planes  of  }  or  45°  through 
these  exterior  lines,  and  finding  by  Prob.  7,  Fig.  9,  their  in- 
tersections with  the  terre-plein,  these  lines  will  be  the  foot 
of  the  barbette  slopes.  A  ramp  having  a  slope  \  leads  from 
.the  terre-plein  to  the  top  of  the  barbette ;  the  width  of  this 
ramp  is  3.30  yds.,  its  interior  line  in  projection  being  on  the 
prolongation  of  the  foot  of  the  banquette  slope.     The  ramp 


16  ONE   PLANE    DESCRIPTIVE   GEOMETRY. 

is  terminated  by  side  slopes  of  },  the  intersections  of  which 
with  the  terre-plein  and  the  slopes  of  the  harbette  and  ban* 
quette  are  found  by  Prob.  7,  Fig.  9.  The  foot  of  the  ramp 
or  its  intersection  with  the  terre-plein  is  also  found  by  the 
same  problem. 

As  the  top  surface  of  the  barbette  is  horizontal,  it  may 
De  necessary  in  some  cases  to  make  the  interior  crest  along 
the  barbette  tlso  horizontal,  in  which  case  the  superior  slope 
of  the  parapet  along  the  barbette  being  higher  than  the 
rest  of  it,  the  two  planes  will  be  connected  by  a  plane  of 
45°,  as  at  C. 

38.  Prob.  19,  PI.  2,  Fig.  4.  To  determine  the  bounding 
surfaces  of  a  ramp  leading  up  an  irregular  surface  and  so 
placed  that  its  axis  or  centre  line  shall  nearly  coincide  with 
the  irregular  surface. 

Let  (8.0),  (9.0),  &c,  be  the  horizontal  curves  of  the  sur- 
face, and  let  a  (8.0)  be  the  point  of  departure  or  foot  of  the 
ramp.  Assuming  the  declivity  of  the  ramp  ±,  for  example, 
from  a,  with  a  radius  of  9  units,  describe  an  arc,  and  join 
by  a  right  bine  the  point  b  where  it  cuts  the  horizontal  (9.0) 
with  the  point.  Repeat  this  construction  from  b  to  c  on  the 
horizontal  (10.0) ;  and  so  on  to  the  top  e  or  point  of  arrival. 
The  broken  line  a-b-c-d-e  will  be  the  projection  of  the  axis. 
But,  to  avoid  the  angular  changes  of  direction,  the  straight 
portions  of  the  axis  may  be  connected  at  the  angular  points, 
by  setting  off  from  b,  for  example,  the  equal  distances  ba ', 
be',  and  connecting  these  points  by  an  arc  of  a  circle  tangent 
to  the  straight  portions.  The  same  construction  being  re- 
peated at  the  other  angular  points,  the  broken  line  will  be 
replaced  by  the  sinuous  line  aa'c',  tfcc,  as  the  axis.  Having 
determined  the  axis,  the  exterior  and  interior  lines  of  the 
top  surface  are  drawn  parallel  to  the  axis,  and  at  a  distance 
from  it  equal  to  half  the  assumed  width  of  the  ramp. 

From  the  position  of  the  axis  the  exterior  half  of  the 
ramp  will  be  in  embankment  and  the  interior  in  excavation. 
To  determine  the  side  slopes  of  the  embankment  pass  planes 
through  the  straight  portions  of  the  exterior  edge  of  the 
ramp,  and  find  by  Prob.  6,  PI.  1,  Fig.  8,  the  horizontals  of 
these  planes,  and  by  Prob.  16,  Fig.  20,  the  intersections  of 
these  planes  with  the  irregular  surface.  The  plane  surfaces  of 
the  side  slopes  thus  determined  are  connected  by  curved  sur- 
faces which  pass  through  the  curved  lines  of  the  exterior  edge. 
These  surfaces  may  be  determined  as  follows :  Take,  for  ex- 
ample, the  point  n  at  the  foot  of  the  plane  side  slope  A  where 
it  cuts  the  radius  through  a'  prolonged,  of  the  arc  a'c' ;  and 
the  point  o  on  the  radius  through  c'  where  it  cuts  the  foot  of 


OXE    PLANE    DESCRIPTIVE    GEOMETRY".  17 

the  plane  side  slope  B.  The  lines  of  which  nv  anil  ou  aro 
the  projections  will  evidently  have  the  same  inclination,  and 
they  may  be  assumed  as  the  lines  of  junction  of  the  plane 
slopes  A  and  B  and  the  curved  side  slope  x.  This  curved 
side  slope  may  then  be  generated  by  the  motion  of  a  right 
line  which  has  the  top  line  of  which  vu  is  the  projection  for 
its  directrix,  whilst  in  its  motion  it  makes  a  constant  angle 
with  the  plane  of  comparison,  and  its  projections  are  con- 
stantly normal  to  the  arc  vu.  From  the  construction  com- 
prising these  conditions  the  foot  no  of  the  curved  portion  x 
of  the  side  slope  is  determined.  The  same  constructions  are 
repeated  to  obtain  the  portions  G  of  the  plane,  and  y,  s  of 
the  curved  side  slopes,  with  the  line  m-n-o-p-q^r-8  the  foot  of 
these  slopes. 

The  side  slopes  of  the  part  in  excavation  A',  B' ,  C  and 
x\  y'  with  the  line  m'-n'-o'-jp'-g'-r'  are  determined  by  like 
constructions. 

The  portions  of  the  top  surfaces  of  the  ramps  bounded 
by  the  arcs  of  circles  are  helicoidal  surfaces,  of  which  the 
axis  is  the  directrix  and  the  plane  of  comparison  the  plane 
director. 

The  curved  surface  side  slopes  are  also  evidently  helicoi- 
dal surfaces,  the  directrices  of  which  are  the  curved  lines 
above  mentioned,  and  the  vertical  lines  through  the  centres 
of  the  arcs  which  are  the  projections  of  those  curved  lines. 

Remarks.  In  the  figure  the  declivity  of  the  side  slopes 
of  the  embankment  is  one-half  the  excavation.  The  decliv- 
ities of  the  curved  portions  of  the  top  are  greater  than  those 
of  the  plane  surfaces,  the  difference  depending  on  the  angle 
between  the  straight  portions  of  the  axis. 

39.  Observations  on  the  best  mode  of  executing 
Drawings. 

Accuracy.  The  first  requisite  in  all  drawings  is  minute 
accuracy,  both  in  the  geometrical  constructions,  and  in  writ- 
ing down  all  letters  and  numbers  which  serve  either  as  ref- 
erences, or  to  give  dimensions.  To  attain  this,  so  far  as 
regards  the  geometrical  part,  judgment  is  to  be  exercised  in 
the  selection  of  the  means  for  establishing;;  on  the  drawing 
the  positions  of  the  various  points  which  are  either  given  or 
to  be  found ;  as  one  method  although  in  theory  as  correct  as 
some  other  may  not,  in  practice,  be  found  to  yield  as  satis- 
factory results.  The  following  remarks  will  serve  to  illus- 
trate this  point : 

1st.  In  setting  off  from,  a  scale  of  equal  parts  several  dis- 
tances, along  a  line,  whether  equal  or  unequal,  the  most  ac- 
curate method  is  to  commence  by  first  setting  off  the  entire 
2 


18  ONE   PLANE   DESCRIPTIVE    GEOMETRY. 

distance,  and  then  the  several  parts ;  talcing  care  to  verify, 
from  the  scale,  the  aggregate  of  the  several  partial  distances ; 
thus  in  the  example  PI.  2,  Fig.  5,  where  the  aggregate  of 
all  the  partial  distances  is  60.33  feet,  commence  by  setting 
off  the  entire  distance  60.33  feet ;  next  50.33,  which  is  the 
sum  of  the  two  distances  20'  and  30'.33,  then  verify  the  re 
maining  10'  by  the  scale. 

2d.  When  a  distance  to  be  set  off  is  so  small  that  it  can- 
not be  laid  down  with  accuracy  by  the  points  of  the  dividers, 
the  following  method  may  be  employed :  set  back,  from  the 
point  from  which  the  required  distance  is  to  be  set  off,  any 
arbitrary  distance,  then  set  forward,  from  this  last  point,  a 
distance  equal  to  the  sum  of  this  arbitrary  distance  and  the 
one  required ;  thus  in  PL  2,  Fig.  6,  where  2'  is  to  be  set  off" 
from  a  towards  c,  set  back  from  a  say  30'  to  b,  then  from  b 
32'  toe. 

3d.  To  set  off  a  point  at  a  given  perpendicular  distance 
from  a  Une,  it  will  mostly  be  found  more  speedy,  and  more 
accurate,  to  take  off  from  the  scale  the  given  distance,  in 
the  dividers,  and,  setting  one  point  on  the  paper,  bring  the 
other  so  that  the  arc  described  by  it,  with  the  given  distance 
as  a  radius,  shall  be  tangent  to  the  line,  than  to  employ  the 
usual  method  of  first  erecting  a  perpendicular  to  the  line 
and  then  setting  off  the  required  point  along  the  perpendic- 
ular; thus  in  Pi.  2,  Fig.  7,  wishing  to  set  off  c  at  20'  from 
ab,  take  20'  in  the  dividers,  and,  by  the  eye,  find  where  one 
point  must  be  placed  so  that  the  other  describing  an  arc  will 
touch  ab.  This  method  will  be  found  convenient  in  drawing 
a  parallel'  to  a  line  at  a  given  distance  from  it  by  setting  off 
another  point  in  the  same  way,  and  drawing  through  the 
two  the  required  parallel. 

4th.  In  setting  off  several  points  for  the  purpose  of  draw- 
ing several  parallels  to  a  given  line,  as,  for  example,  the  par- 
allel lines  which  bound  the  planes  of  a  parapet,  it  will  be 
found  most  speedy  and  accurate  to  draw  first  upon  a  slip  of 
smooth  thin  paper  two  lines  perpendicular  to  each  other, 
then  marking  on  one  of  the  lines  the  respective  given  dis- 
tances of  the  parallels  from  the  other,  and  cutting  the  paper 
close  to  the  line  along  which  the  given  points  are  marked 
off,  so  that  the  strip  when  laid  upon  the  drawing  so  as  to 
have  one  of  its  lines  to  coincide  with  that  to  which  the  par- 
allels are  to  be  drawn,  their  distances  from  it  can  be  pricked 
off  by  a  sharp  pointed  pencil,  or  in  any  other  way.  In  PI. 
2,  Fig.  8,  ab  is  the  line  of  the  drawing ;  A  the  strip  of  paper, 
fc,  fd,  fe,  &c,  the  distances  at  which  the  parallels  are  to  be 
drawn  from  ah,  marked  off  on  the  edge  of  A  perpendicular 


ONE   PLANE   DESCRIPTIVE    GEOMETRY.  10 

to  the  line  f,  which  line  when  A  is  laid  on  the  dn-tving, 
should  coincide  with  ah.  If  the  line  ah  is  somewhat  long,  it 
will  be  better  to  set  oft*  these  points  near  each  of  its  extrem- 
ities and  join  them  by  lines,  than  to  draw  the  parallels  in  the 
usual  way  by  aid  of  the  ruler  and  triangle. 

5th.  When  a  point  is  to  he  constructed  hy  means  of  t/i6 
intersection  of  two  lines  arbitrarily  chosen,  such  a  position 
should  be  assumed  for  the  arbitrary  lines  that  they  shall  not 
form  a  very  acute  angle  at  their  point  of  intersection,  as  in 
that  case  this  point  might  not  be  so  distinct  to  the  eye  as  to 
be  marked  with  accuracy.  For  example,  in  erecting  a  per- 
pendicular to  a  line  at  a  given  point,  and  in  like  problems, 
in  which  points  are  found  by  the  intersections  of  arcs  of  cir- 
cles, it  will  be  best  and  most  convenient  to  take  for  the  radii 
of  the  arcs  the  distance  between  their  centres,  as  the  angle 
between  the  tangents  to  the  arcs  at  their  point  of  intersection 
will  then  be  60°,  which  is  a  sufficient  angle  to  give  accurately 
the  point  where  the  lines  cross.  In  cases  like  Figs.  10,  11, 
Arts.  22,  23,  the  arbitrary  lines  ah,  a'h\  (fee,  should  be  so 
chosen  as  to  intersect  the  horizontals  nearly  at  right  angles, 
and  so,  also,  that  the  resulting  lines,  by  which  the  points  o, 
o' ,  are  determined,  shall  not  intersect  in  too  acute  an  angle. 

In  all  such  cases  of  determining  points,  and  where  a  point 
:'s  pricked  into  the  paper,  it  will  be  found  well  to  designate 
the  point  thus  O,  by  a  small  circle  drawn  around  it  with  the 
lead  pencil,  in  order  that  the  eye  may  see  it  with  more  dis- 
tinctness. 

6th.  In  determining  a  portion  of  a  line  hy  the  construc- 
tion of  two  arbitrary  points,  the  points  should  be  so  chosen 
that  the  portion  required  may  fall  between  them  and  not 
beyond  them.  In  Pi.  1,  Fig.  10,  for  example,  if  the  requir- 
ed portion  of  the  line  of  intersection  of  the  planes  extended 
on  either  side,  beyond  o,  or  o',  or  beyond  both,  the  lines  ah, 
cd,  &c,  should  be  so  chosen  as  to  bring  o  and  o',  as  far  apart, 
at  least,  as  the  length  of  the  required  portion  of  the  line 
which  they  serve  to  determine. 

7th.  No  means  of  verifying  the  accuracy  of  the  construc- 
tion of  points,  or  lines,  should  be  omitted.  In  Pi.  1,  Fig.  9, 
for  example,  other  corresponding  horizontals  should  be  drawn, 
and,  if  the  line  of  intersection  determined  by  the  two  points 
first  found  is  correct,  their  points  of  intersection  also  will 
fall  upon  it.  In  PI.  1,  Figs.  9,  10,  the  scale  of  declivity  of 
the  line  of  intersection  being  determined,  the  references  of 
the  points,  where  it  intersects  the  scales  of  declivity  of  the 
planes,  should  be  the  same  as  the  same  points  on  the  scales, 
if  the  line  has  been  accurately  determined.     A-  general  and 


20  ONE   PLANE   DESCRIPTIVE   GEOMETRY. 

minute  verification  of  all  the  parts  of  the  drawing,  should  be 
made  before  any  portion  of  it  is  put  in  ink. 

Neatness.  This  is  a  not  unimportant  element  in  the  at 
tainment  of  accuracy  in  drawing.  A  few  minutiae,  when 
attended  to,  will  subserve  this  end. 

That  part  of  the  paper  on  which  the  draughtsman  is  not 
working  should  be  kept  covered  with  clean  paper,  pasted  on 
the  edge  of  the  board,  so  as  to  fold  over  the  drawing,  and 
the  parts  which  are  finished  should  be  similarly  protected. 

Before  commencing  the  daily  work  the  paper  should  be 
carefully  dusted,  and  the  scales,  rules  and  triangles  be  care- 
fully wiped  with  a  clean  dry  rag. 

As  few  lines  of  construction  as  possible  should  be  drawn  in 
pencil ;  and  only  that  part  of  each  which  may  be  strictly  ne- 
cessary to  determine  the  point  sought.  As,  for  example,  where 
a  point  is  to  be  found  by  the  intersection  of  two  arcs  of  cir- 
cles ;  when  the  position  of  the  point  can  be  approximately 
judged  of  by  the  eye,  only  a  portion  of  one  arc,  which  will 
embrace  the  point,  may  be  drawn,  and  the  point  where  the 
second  arc  would  intersect  the  first  be  marked  without 
describing  the  arc.  In  PI.  1,  Fig.  10,  instead  of  drawing 
the  entire  lines  ah,  cd,  &c,  it  would  be  simply  necessary  to 
mark  the  points  only  where  they  cut  the  horizontals ;  and, 
in  like  manner,  the  points  o  and  o'  might  be  marked  without 
drawing  the  entire  lines. 

]STo  more  of  any  line  of  the  drawing  should  be  made  in 
pencil  than  what  is  to  remain  permanently  in  ink.  The  ob- 
ject of  these  precautions  is  to  keep  the  paper  from  becoming 
covered  with  dirt  and  the  lines  from  being  defaced  by  the 
wear  of  the  paper. 

Inking.  In  inking  the  lines  the  following  directions  will 
be  found  useful : 

Efface  carefully  all  pencil  lines  that  are  not  to  be  inked; 
and  those  parts  of  the  permanent  lines  which  are  not  to  re- 
main, before  commencing  to  ink. 

When  right  lines  are  tangent  to  curves,  put  in  ink  the 
curve  before  the  right  line ;  draw  all  arcs  of  equal  radii  at 
once,  one  after  the  other ;  if  several  arcs  are  to  be  described 
from  the  same  centre,  it  will  be  well  to  put  a  thin  bit  of  quill 
over  the  point  for  the  end  of  the  dividers  to  rest  on,  to  avoid 
making  a  large  hole  in  the  drawing. 

If  the  drawing  is  not  to  be  colored  with  the  brush,  all 
the  lines  of  one  color  should  be  put  in  before  commencing 
on  those  of  another. 

If  one  of  the  bounding  lines  of  a  surface  is  to  be  made 
heavier  than  the  others,  its  breadth  should  be  taken  from 


TlaM'l. 


Tig.  14. 


;  '°oj 


■-■-----  — -  -(9.0) 

!  *•  ^  '"  - 

" 


ONE    PLANE    DESCRIPTIVE   OEOMETBY.  21 

the  surface  they  limit  and  not  be  added  to  it ;  aid  when  the 
heavy  line  forms  the  boundary  of  two  surfaces,  its  breadth 
must  be  taken  from  the  one  of  greatest  declivity. 

Coloring.  "When  the  drawing  is  to  be  colored,  all  lines 
that  are  not  to  be  black  may  be  put  in  first  with  black,  mak- 
ing them  very  faint,  so  that  they  may  receive  their  appro- 
priate colors  after  the  drawing  is  otherwise  completed. 

]STo  heavy  line  should  be  put  in  until  the  work  with  the 
brush  is  completed. 

When  all  the  lines  are  in,  the  drawing  should  be  thor- 
oughly cleaned  with  stale  bread-crumb ;  and  then  have  sev- 
eral pitchers  of  water  dashed  over  it,  the  board  being  placed 
in  an  inclined  position  to  allow  the  water,  colored  by  the  ink 
lines,  to  escape  rapidly,  and  not  to  discolor  the  paper. 

In  using  the  brush,  whether  for  flat  tints,  or  graded,  the 
requisite  depth  of  tint  should  be  reached  by  a  number  of 
faint  tints  laid  over  each  other ;  this  is  especially  necessary 
in  laying  tints  of  blacks,  browns,  and  reds. 

To  obtain  an  even  flat,  or  graded  tint,  on  dry  paper  re- 
quires considerable  skill.  The  best  plan  for  this  is,  first  to 
wet  with  a  large  brush,  or  clean  rag,  the  surface  on  which 
the  tint  is  to  be  laid,  then,  with  a  slightly  moist  rag,  clear 
the  surface  of  water,  and  before  the  paper  has  time  to  dry 
to  lay  on  the  tint.  With  this  precaution,  the  heaviest  tints 
of  Chinese  ink,  the  most  difficult  of  all  to  manage  on  dry 
paper,  can  be  neatly  laid  down. 

Titles,  &c.  The  lettering  and  numbering  of  a  drawing 
should  be  in  ordinary  printed  character ;  this  is  particularly 
requisite  in  the  numbering,  to  avoid  misapprehensions  which 
often  arise  from  individual  peculiarities  in  writing  numbers. 

As  has  been  already  remarked,  references  are  written  in 
black,  within  brackets  which,  when  practicable,  embrace  the 
point  referred  to.  When  not  practicable,  a  small  dotted  line 
may  lead  from  the  point  to  the  reference ;  thus,  0. .  .(25.50) ; 
but  to  distinguish  references  from  other  numbers  the  desig- 
nation of  the  unit  is  omitted. 

All  horizontal  distances  between  points  are  written  upon 
a  dotted  line  drawn  between  the  points,  with  an  arrow-head 
at  each  end ;  where  several  partial  distances  in  a  right  line 
are  marked,  it  will  be  also  well  to  mark  the  total  distance : 
the  latter  may  be  written  above  or  beneath  the  former,  PI. 
2,  Fig.  5.  ^ 

In  writing  horizontal  distances,  the  usual  designation  of 
the  unit  is  always  written  thus,  y  for  yards,  '  for  feet,  (tec. 
All  the  numbers  must  be  expressed  in  the  same  unit ;  the 
fractional  parts  being  in  decimals. 


22  ONE   PLANE   DESCRIPTIVE   GEOMETRY. 

References  and  horizontal  distances  cannot  be  too  miu  h 
multiplied,  in  order  to  avoid  misapprehensions,  and  the  re- 
sults of  errors  of  construction,  as  well  as  to  save  the  time 
that  would  be  taken  in  applying  dividers  to  the  drawing  to 
find  from  the  scale  affixed  to  it  the  dimensions  of  any  part. 

Scale.  A  scale  very  accurately  constructed  should  bo 
affixed  to  the  drawing  before  it  is  cut  from  the  board ;  so 
that  the  shrinkage  of  the  paper,  which  is  about  5^5  may 
affect  all  the  parts  equally,  and  the  scale  thus  be  made  to 
correspond  to  the  real  lengths  of  the  lines  on  the  drawing. 
The  scale  should  be  divided  according  to  the  decimal  system, 
as  being  most  convenient  for  counting  off. 

The  first  division  of  the  scale  should  furnish  the  units 
and  also  their  decimal  parts,  if  the  scale  bears  that  propor- 
tion to  the  true  dimensions  of  the  object  represented  which 
will  admit  of  these  divisions.  This  first  division  is  numbered 
from  right  to  left,  PI.  2,  Fig.  9,  the  zero  point  being  on  the 
right,  the  10  point  on  the  left ;  the  succeeding  divisions,  to 
50  inclusive,  should  each  be  equal  to  the  first  division,  con- 
taining ten  units  each.  The  remaining  divisions  may  con- 
tain fifty  units  each.  It  will  be  seen  that  any  number  of 
tens,  units,  or  fractional  parts  of  a  nnit  can  thus  be  readily 
taken  off  from  the  scale  by  the  dividers.  The  scale  should 
be  long  enough  to  give  the  dimensions  of  the  longest  line  on 
the  drawing. 

The  proportion  which  the  scale  bears  to  the  true  dimen- 
sions of  the  object  should  be  written  above  the  scale ;  thus, 
Scale  one  inch  to  ten  yards,  or  3  }q.  And  the  designation 
of  the  unit  of  the  drawing  should  be  annexed  to  the  last 
division  on  the  scale,  as  yds.  for  yards,  ft.  for  feet,  etc. 

Note. — For  more  detailed  directions  on  the  mechanical 
or  instrumental  methods  of  geometrical  drawing,  see  Mahan'a 
Industrial  Drawing. 


Hale  2. 


STONE   CUTTING.  ^3 


STONE  CUTTING. 

The  object  of  tins  Article  is  to  explain  the  geometrica 
methods  of  representing  the  more  usual  and  elementary 
combinations  of  blocks  of  stone -in  walls  and  arches,  by 
means  of  their  projections ;  and  from  these  and  the  data  of 
the  problem  to  deduce  the  true  dimensions  of  the  bounding 
surfaces  and  lines  of  each  part. 

Walls  bounded  by  Plane  Surfaces.  In  walls  of  cut 
stone a  the  blocks  are  usually  separated  by  horizontal  and 
vertical  joints;  the  latter  being  in  vertical  planes  perpendic- 
ular to  the  face  of  the  walls,  and  which  are  termed  plane* 
of  right  section,  to  distinguish  them  from  other  vertical 
planes  of  section.  When  the  face  of  the  wall  is  inclined  to 
the  horizon,  its  slope,  or  batir,  is  usually  expressed  by  the 
ratio  of  the  base  of  the  slope  to  the  perpendicular,  measured 
in  the  plane  of  right  section ;  or  the  slope  is  said  to  be  so 
many  base  to  so  many  perpendicular.  In  the  right  section 
A' It' CD',  (PI.  A,  Fig.  1,)  for  example,  the  inclination  of 
the  face  A'C  to  the  base  of  the  wall  A  B',  is  measured  by 
dividing  the  perpendicular  C  E'  from  C  upon  the  line  A'B\ 
by  the  distance  A E'  between  the  point  A'  and  the  foot  of 
the  perpendicular.  The  quotient  ^',  thus  obtained,  is  ev- 
idently the  natural  tangent  of  the  angle  C'A'E' ;  and  the 
most  convenient  method  of  representing  the  batir  is  by  a 
fraction ;  the  numerator  expressing  the  number  of  units  in 
the  perpendicular,  and  the  denominator  the  corresponding 
number  of  units  in  the  base ;  thus  a  batir  of  f  expresses  a 
slope  of  six  perpendicular  to  one  base ;  a  batir  of  f ,  one  of 
three  perpendicular  to  two  base,  &c. 

Prob.  1.  Having  given  the  right  section  of  a  wall,  to 
construct  the  projections  of  its  hounding  lines,  and  the  edges 
of  the  horizontal  and  vertical  joints. 

Let  A'B'D'C,  (PI.  A,  Fig.  1,)  be  the  right  section  ;  the 
base  A'B'  and  the  top  CD'  being  horizontal ;  the  face  A'C 
having  a  batir  —  =  f ;  and  the  back  B'D'  vertical. 

Draw  a  lino  A  A, ,  to  represent  the  foot  of  the  face  in  pl?n. 
Parallel  to  A  A,  draw  CO, ,  and  at  a  distance  from  it  eq/ial  xo 

•  See  Mahau's  Civil  Engineering,  Art.  351. 


24  STONE   CUTTING. 

A ' E\  the  distance  of  the  point  A'  from  the  foot  of  the  per 
pendicular  drawn  from  C  in  the  plane  of  right  section ;  CC 
will  be  the  top  line  of  the  face  in  plan.  Parallel  to  CCI ,  and 
at  a  distance  CD  ,  the  breadth  of  the  wall  at  top,  draw 
BBt  ,  which  will  be  the  projection  of  the  hack. 

At  any  convenient  distance  from  BB '  draw  A" A'"  par- 
allel to  it,  to  represent  the  foot  of  the  wall  in  elevation;  the 
top  C"D"  will  be  drawn  parallel  to  A" A'",  and  at  the 
height  BD'  of  the  top  above  the  base. 

To  draw  the  projections  of  the  horizontal  edges,  suppose 
the  wall  divided  into  four  equal  courses  by  horizontal  joints. 
As  the  batir  of  the  wall  is  f ,  the  base  of  the  slope  of  each 
of  these  equal  courses  will  be  one-sixth  of  its  height;  if 
•lines  therefore  are  drawn  parallel  to  the  foot  AA/ ,  and  at  a 
distance  from  each  other  equal  to  $  the  height  of  each  course, 
these  lines  will  be  the  projections  in  plan  of  the  horizontal 
edges,  as  shown  on  the  right  of  the  plan.  As  the  projections 
of  the  vertical  edges  are  contained  in  planes  of  right  section, 
they  will  be  drawn  perpendicular  to  the  horizontal  ones, 
and  breaking  joints  with  them ;  as  represented  on  the  same 
portion  of  the  plan. 

The  horizontal  edges  in  elevation  will  be  drawn  parallel 
to  A"  A'" ,  and  at  a  distance  from  each  other  equal  to  the 
height  of  a  course.  The  vertical  edges  will  be  drawn  per- 
pendicular to  these  last,  and  corresponding  to  their  projec- 
tions in  plan. 

Remark.  If  the  projections  of  any  horizontal  line  of 
the  face,  at  a  given  height  above  the  foot  of  the  wall,  are 
required,  as  mn  and  m"n',  for  example,  it  is  evident  that 
m"n'  will  be  drawn  in  elevation  at  the  given  height  above 
the  foot  A"  A'",  and  that  mn  will  be  parallel  to  AA/  in  plan, 
and  at  a  distance  from  it  equal  to  }  the  height  of  m"n'. 

JProb.  2.  Having  given  the  batir  of  the  faces  of  two 
walls  that  intersect,  the  foot  of  each  being  in  the  same  hori- 
zontal plane,  to  draw  the  projections  of  the  line  of  intersec- 
tion of  the  faces. 

Let  Aa,  {Fig.  1,)  be  the  foot  of  one  wall  in  plan,  and  the 
batir  of  its  face  f ;  ah  the  foot  of  the  other,  and  the  batir  of 
its  face  \. 

It  is  evident  that  the  point  a  will  be  one  point  of  the 
intersection  in  plan.  If  now  a  horizontal  line  be  drawn  in 
each  face  at  the  same  altitude,  they  will  intersect  and  give  a 
second  point  of  the  intersection  of  the  faces.  Assuming  any 
altitude  for  the  horizontal  line  on  one  face,  its  projection 
mn,  in  plan,  will  be  parallel  to  Aa,  and  at  }  of  the  assumed 
altitude  from  it.     In  like  manner,  the  projection  no  of  the 


6T0XE   CUTTING.  25 

corresponding  line  on  the  other  face  having  the  batir  f  will 
be  parallel  to  ah,  and  at  £  the  assumed  altitude  from  it. 
Drawing,  therefore,  a  line  through  the  points  a  and  n,  this 
will  be  the  intersection  in  plan. 

To  obtain  the  intersection  in  elevation,  draw  the  foot  of 
the  wall  A"  a'  in  elevation,  also  the  line  m"n'  at  the  assumed 
altitude.  The  point  a  will  be  projected  in  a',  and  the  point 
n  in  n ';  and  the  line  a'n' ,  drawn  through  them,  will  be  the 
intersection  in  elevation. 

Proh.  3.  To  construct  the  projections  of  the  hounding 
Lines  and  edges  of  the  joints  of  a  huttress  against  the  inclin- 
ed face  of  a  given  wall  ;  the  hase  of  the  huttress  heing  given, 
and  heing  in  the  same  horizontal  plane  as  the  hase  of  the 
wall;  the  faces  of  the  huttress,  its  end,  and  the  top  to  have 
given  slopes. 

Along  the  foot  AAt  set  off  the  breadth  of  the  buttress 
ad  at  its  base,  and  construct  the  two  sides  ah,  cd;  and  the 
end  he  of  the  base.  Let  the  batir  of  the  face  of  the  wall 
foe  \ ;  that  of  the  two  faces  and  the  end  of  the  buttress  f ; 
and  that  of  its  top  }. 

By  Prob.  2  construct  the  projections,  in  plan  and  eleva- 
tion, of  the  intersections  ae,  a'e ,  and  dh,  d'h',  of  the  faces 
of  the  buttress  and  wall ;  and  those  hf  hf,  and  eg,  c'g',  of 
the  end  and  faces. 

To  construct  the  projections  of  the  top  surface  of  the 
huttress. 

Suppose  that  the  top  line  of  the  buttress  eh,  e'h',  where 
it  joins  the  wall,  is  of  the  same  altitude  as  the  top  of  the 
wall ;  and  that  the  top  surface  from  this  line  outwards  from 
the  wall  has  the  given  slope  j.  Now,  if  a  line  as  y'zL 
be  drawn  parallel  to  e'h'  the  top  line,  and  at  any  assumed 
distance  below  it,  this  line  may  be  regarded  as  the  projection, 
of  a  horizontal  line  in  the  top  surface  of  the  buttress ;  and 
its  corresponding  projection  in  plan  will  be  a  line  yz  parallel 
to  eh,  and  at  six  times  the  distance  from  it  that  y'z'  is  below 
e'h'.  Having  drawn  these  two  lines  of  indefinite  length, 
construct  the  projections  of  the  horizontal  in  the  face  of  the 
buttress  which  is  at  the  same  distance  below  the  top.  The 
projection  in  elevation  will  be  a  continuation  of  the  same 
line  y'z',  and  in  plan  its  projection  will  be  parallel  to  the 
foot  ah,  and  at  a  distance  from  it  equal  to  \  its  altitude 
*bove  it.  Drawing  an  indefinite  line  xy  parallel  to  ah  at 
this  distance  from  it,  the  point  y,  where  it  intersects  the  line 
yz,  will  be  a  point  of  the  projection  in  plan  of  the  intersec- 
tion of  the  top  surface  and  face  of  the  buttress.  The  point 
e.  is  another  point;  joining  e  and  y  by  a  line  and  prolonging 


26  STONE   CUTTING. 

it  unt.l  it  intersects  the  projection  of  the  intersection  of  the 
face  and  end  at  f,  this  line  ef  will  be  the  projection  of  one 
of  the  top  surface  in  plan.  The  other  side  hzg  will  be 
found  by  a  like  construction.  The  points  /  and  g  being 
joined  will  be  the  projection  of  the  end  of  the  top  surface. 

To  obtain  the  corresponding  lines  in  elevation;  the  points 
?/  and  z  are  projected  into  //'  and  z' \  the  points  e',  y1,  and  h', 
z\  are  joined  by  lines  which  are  prolonged  to  meet  the  lines 
which  are  the  projections  of  the  exterior  edges  of  the  hut- 
tress  at/'  and  g',  which  correspond  to  /and  g  in  plan,  and 
the  points  /'  and  g'  are  joined  ;  e'f'g'ti  is  the  elevation  of 
the  top  surface. 

The  projections  of  the  edges  of  the  vertical  joints  in  plan 
will  be  perpendicular  respectively  to  the  lines  be  and  cd,  and 
breaking  joints  as  shown  on  the  right  portion  of  the  plan. 
The  projections  of  these  lines  in  elevation  will  be  found  by 
projecting  their  extremities  into  the  corresponding  projec- 
tions of  the  horizontal  edges  in  elevation,  as  shown  on  the 
elevation  of  the  face  and  a  portion  of  the  end,  on  the  right. 

Prob.  3,  Case  2  (PL  B,  Figs.  1,  2,  3).  Having  given  the 
cross  section  of  a  brook,  or  other  smaU  natural  water  wag, 
over  which  a  full  centre  archedstone  culvert  is  to  be  thrown, 
to  support  an  embankment  of  a  roadway,  of  a  given  height 
above  the  natural  surface  of  the  ground,  to  construct  the 
bounding  lines  of  a  wing  wall  with  plane  bounding  sur- 
faces. 

Figs.  1,  2  are  the  elevation  and  plan,  or  the  vertical  and 
horizontal  projections  of  the  parts  ;  P  Q  being  the  ground 
line.  Fig.  3  is  a  section  and  elevation,  through  the  axis  of 
the  arch,  on  the  vertical  plane  of  which  11  S  is  the  ground 
line.  M,  M'  are  the  slopes  of  the  embankment,  jST,  N'  the 
bottom  of  the  brook. 

Let  C  B  Z',  Fig.  l,be  the  cross  section  of  the  side  bank 
of  the  water  way,  and  of  the  adjacent  level  ground;  0  the 
centre  of  the  semicircle  of  the  full  centre  arch,  taken  on  the 
level  C  B  of  the  natural  surface;  L  I  the  level  of  the  top  of 
the  embankment;  1/  E',  Fig.  2,  the  foot  of  the  embankment 

The  wing  wall  and  arch  {Figs.  1,  2),  are  supported  upon 
a  general  substructure,  the  height  of  which  is  A  A';  the 
plan  of  that  portion  of  which,  supporting  the  wing  wall,  is 
shown  by  A'  B'  C.  The  faces  of  this  substructure  being 
vertical,  and  projecting  a  distance,  represented  by  Z  A,  be- 
yond the  springing  line  of  the  arch,  the  foot  of  the  wing  wall, 
and  the  foot  of  the  embankment;  the  point  B'  being  taken 
on  the  crest  B'  K  of  the  side  bank. 

Through   the  line  Z  D,  Z'  D',  the  foot  of  the  wing  wall, 


STONE    CUTTING.  27 

the  plane  of  its  face  is  passed;  the  top  point  of  which  X,  X' 
is  found  by  drawing  a  line  m  n  parallel  to  Z'  D',  and  at  a 
distance  from  it  equal  to  the  base  of  the  batir  corresponding 
to  the  height  of  the  embankment  L  I  above  D  Z,  and  taking 
its  intersection  X,  X'  with  the  top  line  of  the  head  of  the 
arch.  Joining  X  Z,  it  will  be  the  elevation  of  the  intersec- 
tion of  the  face  of  the  wing  wall  with  the  end  of  the  arch. 

The  top  surface  of  the  wing  wall  X."  T>"  (Fig.  8),  receives 
the  same  slope  as  the  side  slope  of  the  embankment  M,  M', 
and  is  here  taken  to  coincide  with  it.  The  wing  wall  is  ter- 
minated at  the  end  by  a  vertical  plane  D"  F"  parallel  to  the 
head  wall  of  the  arch.  The  thickness  I  X,  FX'  of  the  wing 
wall  at  top  is  assumed. 

Joining  the  points  X'  D'  and  X  D,  the  interior  edge  of 
the  top  line  of  the  wing  wall  is  found.  Drawing  I  H  and 
I'  IF  respectively  parallel  to  these,  the  exterior  lines  are 
found. 

The  lower  end  of  the  wing  wall  is  terminated  by  what  is 
termed  a  newel  stone,  which  serves,  in  this  case,  as  a  buttress. 
The  height  of  this  stone  D"  F"  is  arbitrary,  as  is  also  its 
slope  F"  G"  on  top.  Assuming  these,  the  intersection  of 
the  vertical  plane,  terminating  the  wing  wall  with  its  face, 
will  be  the  lines  F'  D',  F  D,  parallel  respectively  to  X'  Z' 
and  X  Z.  The  lines  F'  G',  F  G,  and  H'  E',  H  E,  which  also 
are  parallel,  will  be  found  by  Prob.  3. 

The  vertical  joints  of  the  face  of  the  wing  wall  are  per- 
pendicular to  its  face.  Drawing  the  line  X'  Y'  perpendicular 
to  D'  Z',  and  its  corresponding  projections  X  Y,  X''  Y"  on 
Figs.  1,  3,  the  directions  of  the  edges  of  the  vertical  joints, 
as  x'  y\  x  y,  x"  y",  will  be  parallel,  on  their  respective  Figs., 
to  these  lines. 

The  top  of  the  wing  wall,  instead  of  a  coping,  is  formed 
with  elbow  joints  uniting  with  the  horizontal  joints.  The 
portion  of  the  joint  forming  the  elbow  is  perpendicular  to 
the  top  surface.  Drawing  then  a  line  Z"  W"  (Fig.  3),  per- 
pendicular to  D"  X",  and  its  corresponding  projections  Z  W, 
Z'  W  on  Figs.  1,  2,  these  will  be  the  directions  z  w,  z'  w', 
z"  w",  of  the  elbows.  The  depth  of  the  elbow  is  arbitrarily 
assumed,  by  drawing  a  line  n"  q"  on  Fig.  3  parallel  to 
D"  X". 


28  STONE   CUTTING. 


CYLINDRICAL 


AND  OTHER 


A.H  O  H  E  S. 


To  facilitate  the  geometrical  operations  for  determining 
the  bounding  surfaces  and  lines  of  the  voussoirs  of  arches,  a 
few  preliminary  problems  and  theorems,  on  which  these 
operations  are  based,  will  first  be  explained. 

Prob.  4,  {PI.  A,  Fig.  2.)  Saving  given  a  semi  cylinder, 
the  right  section  of  which  is  a  semicircle,  and  its  axis  and 
two  bounding  elements  being  horizontal,  to  construct  the  pro- 
jections of  the  intersection  of  the  cylinder  by  a  plane  inclined 
to  its  axis  and  having  a  given  inclination  to  the  horizontal 
plane  containing  the  axis  /  also,  the  projection  of  the  inter- 
section of  this  semi  cylinder  with  another  semi  cylinder  with 
a  semicircle  also  for  its  right  section,  the  axis  and  bounding 
elements  of  this  last  being  in  the  same  horizontal  plane  as 
those  of  the  first ;  and  then  to  develop  the  portion  of  the 
first  semi  cylinder  which  lies  between  the  given  plane  and 
the  other  cylinder. 

Let  a'c'b'  be  the  right  section  of  the  given  cylinder,  and 
o'  its  centre ;  the  line  a'V  being  horizontal.  Let  a' A  and 
b'B  be  the  horizontal  projections  of  its  bounding  elements, 
and  o'G  that  of  its  axis.  Let  ab  be  the  trace  of  the  given 
inclined  plane  on  the  horizontal  plane  of  the  bounding  ele- 
ments ;  AB  one  of  the  bounding  elements  of  the  other  cylin- 
der, and  LM  its  axis.  The  quadrant  AL'  the  half  of  the 
right  section  of  this  cylinder ;  L  the  centre  of  this  quadrant. 
°  1st.  Taking  any  two  elements  of  the  given  cylinder,  at 
the  same  height,  as  x'x^  and  yfyt,  above  a'b',  they  will  be 
projected  in  plan  parallel  to  the  axis  o'C\  and  will  be  drawn 


STONE   CUTTING.  29 

indefinitely  through  the  points  a?2  and  yv  The  given  ineli  ned 
plane  will  cut  these  elements  at  the  same  height  x'x„  and  if 
the  projection  xy  of  a  horizontal  line  in  this  plane,  at  the 
height  x'x„  be  drawn,  the  points  x  and  y,  where  it  cuts  the 
two  elements  of  the  cylinder,  will  be  two  points  of  the  re- 
quired projection  in  plan.  To  construct  this  line  xy,  let  the 
given  inclination  of  the  plane  be  \ ;  the  projection  of  this 
horizontal  line,  which  is  at  the  height  x'xt  above  the  foot  ab 
of  the  plane,  will  be  {Prob.  2)  parallel  to  ab,  and  at  a  distance 
from  it  equal  to  \  of  x'xt ;  drawing  therefore  xy  parallel  to 
ab,  and  at  this  distance,  it  will  be  the  required  projection  in 
plan.  The  points  x  and  y  thus  found  will  be  two  points  of 
the  projection  in  plan  required.  In  the  same  way  any  num- 
ber of  points  can  be  found,  and  the  curve  axeyb,  traced 
through  them,  will  be  the  required  projection  in  plan. 

The  construction  just  explained,  although  very  simple, 
may  be  abridged  as  follows :  Through  a'  draw  a'f'  perpen- 
dicular to  a'b' ;  prolong  y'x'  to  the  left,  and  set  off  from  m'", 
where  it  cuts  a'f',  the  distance  m'"x'"  equal  to  mx,  as  before 
found.  Through  a'x'"  draw  the  indefinite  line  a'e.  Now, 
to  construct  the  projection  of  any  other  point  in  plan,  as  c 
on  the  element  at  the  height  d ;  through  d  draAV  a  line  par- 
allel to  a'b',  take  the  part  o'"d"  intercepted  between  a'f  and 
a'e'  and  set  it  off  from  o,  where  the  projection  of  the  ele- 
ment through  d  cuts  ab,  to  c  along  the  projection  of  the  ele- 
ment; c  will  be  the  required  point.  This  is  evident  from 
the  relations  which  the  heights  and  horizontal  distances  con- 
sidered bear  to  each  other. 

2d.  To  find  the  projection  in  plan  of  the  intersection  of 
the  cylinders.  Draw  AD  perpendicular  to  AL.  If  a  dis- 
tance Ar"  equal  x'x3  is  set  off  on  this  line,  and  a  parallel  to 
AL  be  drawn  through  r",  the  point  u"  where  it  cutis  the 
quadrant  will  give  the  point  on  it  through  which  the  element 
of  the  second  cylinder,  at  the  height  x'xi  of  the  two  elements 
at  x'  and  y',  is  drawn.  Through  u"  drawing  an  indefinite  line 
parallel  to  AB,  the  bounding  element  of  the  second  cylin- 
der, it  will  be  the  projection  in  plan  of  the  element  at  the 
height  Ar"=x'xi ;  and  the  points  w  and  v,  where  it  cuts  the 
two  projections  of  the  elements  of  the  first  cylinder  at  the 
same  height,  will  be  two  points  of  the  required  projection. 
In  the  same  way  other  points  would  be  found,  by  construct- 
ing the  projections  in  plan  of  corresponding  elements  on 
the  two  cylinders. 

This  operation,  like  the  former,  may  be  also  abridged  aa 
follows :  Through  V  draw  a  perpendicular  b'd  to  a'b'.  With 
a  radius  equal  to  AL  describe  a  quadrant  tangent  to  I'd'  at 


30  stoxe  crrrrs-G. 

b'.  Now,  if  a'y'  be  prolonged  to  the  right,  it  is  evident  that 
the  distance  r'"u'",  intercepted  between  bd  and  the  quad- 
rant, is  equal  to  r"vl\  or  to  rw.  In  like  manner,  the  point 
G  in  plan  is  obtained  by  setting  off  from  t,  on  the  element 
-4_Z?and  along  oC,  the  distance  W=HC.  The  curve  AmOoB^ 
drawn  through  the  points  thus  determined,  will  be  the  pro- 
jection in  plan  of  the  intersection  of  the  two  cylinders. 

3d.  To  make  the  development  of  the  portion  of  the  cyl- 
inder which  lies  between  the  two  intersections  thus  deter- 
mined, it  will  be  necessary  to  obtain  the  distances  of  the 
points  of  these  two  intersections  from  a  curve  of  right  sec- 
tion ;  since  the  tangent  to  this  curve  at  any  point  being  per- 
pendicular to  the  element  of  the  cylinder  at  the  same  point, 
the  curve  when  developed  will  also  be  perpendicular  to  the 
elements  when  developed,  and  will  therefore  develop  into  a 
right  line. 

To  determine  the  relative  positions  of  these  curves  in 
development ;  first  develop  the  curve  of  right  section  a'c'b', 
on  the  line  a'b'  prolonged  to  the  right,  by  setting  off  the  dis- 
tances b'y",  b'c",  &c,  to  a",  equal  respectively  to  the  lengths 
of  the  arcs  b'y',  b'c',  &c,  to  a'.  Through  the  points  y",  c" , 
dec,  draw  lines  perpendicular  to  b'a" ;  these  will  be  the  de- 
veloped elements  of  the  cylinder  through  y',  c,  &c  Set 
off  along  these  lines  the  distances y"v/ ,  c"Ot ,  <kc,  respectively 
equal  to  the  distances  y„v,  o'C,  &c,  and  through  the  points 
jd,  v„  C„  wx,  Ax,  draw  a  curve.  This  is  the  developed  inter- 
section of  the  two  cylinders.  Make  the  same  constructions, 
on  the  same  developed  elements,  with  respect  to  the  distances 
of  the  points  y,  c,  x,  a,  from  a'b' ;  the  curve  &?,«„  drawn 
through  these  points,  will  be  the  developed  intersection  of 
the  oblique  inclined  plane  and  the  cylinder. 

In  like  manner,  if  the  given  cylinder  were  cut  by  a  plane 
perpendicular  to  the  horizontal  plane  and  oblique  to  its  axis, 
of  which  ab,  for  example,  is  the  trace,  the  developed  curve 
of  its  intersection  would  be  obtained  by  setting  off  along  the 
developed  elements  the  distances  of  the  points  a,  m,  o,  cVrc, 
from  a'b',  and  through  the  points  thus  determined  drawing 
a  curve  boxaim 

Rema/rk.  The  curve  of  right  section  in  development 
serves  only  as  a  fixed  line  from  which  the  relative  positions 
of  other  points,  with  respect  to  it  and  to  each  other,  can  be 
determined;  since  it  develops  into  a  right  line,  and  the 
elements  in  development  are  perpendicular  to  it.  The  posi- 
tion of  this  curve  may  be  therefore  fixed  arbitrarily,  as  maj 
be  found  most  convenient  for  the  purposes  of  the  drawing. 


STONE   CUTTING.  31 

Prob.  5,  (PI  A,  Fig.  4).  Let  the  semicircle  A'C'B'  be 
tLe  curve  of  intersection  of  a  given  semi  cylinder  by  a  ver- 
tical plane,  the  diameter  AB1  being  horizontal,  and  suppose 
the  bounding  elements  through  A'  and  B'  to  be  limited  at 
a  horizontal  plane  at  the  distance  A  A  below  A'B',  and  to 
pierce  this  plane  at  the  points  E  and  D,  on  a  line  parallel 
to  AB  and  at  a  given  distance  from  it ;  these  elements  being 
oblique  both  to  the  vertical  plane  and  to  the  horizontal  plane, 
and  therefore  not  projected  on  either  into  their  true  lengths, 

Let  DM  be  the  axis,  and  ED  a  bounding  element  of 
another  semi  cylinder,  the  right  section  of  -which  is  a  semi- 
circle; L2£  and  ED  being  also  in  the  given  horizontal 
plane. 

It  is  proposed,  with  these  data,  to  find  the  lengths  of  the 
elements  of  the  obit 'que 'cylinder  intercepted  between  the  ver- 
tical plane  of  ABC'  and  the  horizontal  cylinder;  the  curve 
of  right  section  of  the  oblique  semi  cylinder  at  any  assumed, 
point;  and  the  development  of  the  portion  of  it  which  lies 
between-  the  given  vertical  plane  and  the  horizontal  semi  cyl- 
inder. 

1st.  The  simplest  method  of  finding  the  true  lengths  of 
the  elements  between  the  vertical  plane  and  horizontal  cyl- 
inder will  be  to  construct  their  projections  on  another  vertical 
plane  parallel  to  them.  Let  BN  be  the  trace  of  such  a  plane 
on  the  horizontal  plane  containing  the  axis  LM  of  the  hori- 
zontal semi  cylinder,  and  Be'  its  trace  on  the  given  vertical 
plane.  This  assumed  plane  cuts  from  the  horizontal  semi 
cylinder  an  ellipse  of  which  DN  is  evidently  the  semi  trans- 
verse axis,  and  the  radius  of  the  semi  cylinder,  which  is 
equal  to  the  distance  between  the  axis  and  the  bounding 
element  ED,  is  the  semi  conjugate ;  setting  off  this  distance 
from  N  to  N\  on  a  perpendicular  to  DN,  and  describing 
the  quadrant  of  an  ellipse  DN' ,  on  these  lines  as  semi  axes, 
it  will  be  the  half  of  the  curve  cut  from  the  semi  cylinder, 
and  will  be  its  position  when  the  plane  in  which  it  lies  is 
revolved  around  its  trace  BN  to  coincide  with  the  given 
horizontal  plane.  In  this  revolved  position  of  the  plane,  the 
line  Be',  in  which  it  cuts  the  given  vertical  plane,  will  be 
found  in  BO"  perpendicular  to  BN.  As  this  plane  contains 
the  bounding  element  of  the  oblique  semi  cylinder  projected 
in  BD,  this  element  will  be  found  in  DB '  when  revolved, 
the  height  BB"  being  equal  to  BB',  the  height  above  AB 
in  which  the  element  pierces  the  given  vertical  plane. 

jSTow,  if  any  other  vertical  plane  be  passed  parallel  to 
the  one  assumed,  as  that  of  which  OG/ ,  and  CC',  parallel 
respectively  to  BN  and  Be'  are  the  traces,  it  will  cut  from 


32  STOXE    CUTTING. 

the  horizontal  semi  sylinder  an  ellipse,  equal  to  the  one 
already  described;  and  from  the  given  vertical  plane  the 
line  CC ;  and.  as  it  contains  also  the  element  projected  in 
CC\,  by  revolving  this  plane  also,  like  the  last,  tracing  the 
quadrant  of  the  ellipse  cut  out,  the  line  CC\  and  the  element 
in  their  revolved  positions;  the  portion  of  this  element  be- 
tween the  vertical  plane  and  the  horizontal  cylinder  may 
thus  be  determined.  In  like  manner,  the  corresponding 
lengths  of  any  other  elements,  as  those  which  are  projected 
in  mm/ ,  and  nn/ ,  might  be  found.  But  as  these  successive 
operations  would  be  long,  a  more  simple  and  expeditious 
method  is  resorted  to,  as  follows:  As  the  elements  of  the 
horizontal  semi  cylinder  are  parallel  to  the  given  vertical 
plane,  if  all  the  points  on  this  plane  and  cylinder  are  pro- 
jected on  the  assumed  vertical  plane  of  which  BS  and  Be' 
are  the  traces,  by  a  system  of  lines  oblique  to  this  plane  and 
parallel  to  the  elements  of  the  horizontal  cylinder,  it  is  evi- 
dent that  all  the  ellipses  cut  from  the  horizontal  cylinder 
will  be  projected  into  the  one  cut  from  it  by  the  assumed 
vertical  plane;  that  all  the  lines,  as  CC,  mm'.  .  ■•-■'.  &c,  cut 
from  the  given  vertical  plane,  will  be  projected  in  Be' ;  and, 
in  the  revolved  position  of  the  assumed  vertical  plane,  will 
be  found  in  BC" ;  whilst  the  portions  of  elements  of  the 
oblique  semi  cylinder,  which  lie  between  the  horizontal  semi 
cylinder  and  the  given  vertical  plane,  will  be  projected  on 
the  assumed  vertical  plane  in  their  true  lengths,  and,  in  its 
revolved  position,  will  be  found  parallel  to  B'D,  and  drawn 
through  points  n",  C",  &c,  at  the  same  height  above  B,  on 
the  line  BC",  as  the  corresponding  points  ml,  <  ,  n',  are 
above  AB.  Drawing  these  parallels,  the  portions  n'"m  , 
CO'",  &c,  between  the  line  BC"  and  the  curve  1)X',  will 
be  the  lengths  required. 

As  there  are  two  elements  on  the  oblique  semi  cylinder, 
one  on  each  side  of  the  highest  one,  projected  in  CC, ,  as 
those  projected  in  mm, ,  and  nnt ,  which  are  of  tlte  same  alti- 
tude, the  lines  B"D,  n'"m",  Arc,  will  be  respectively  the 
revolved  positions  of  the  projections  of  the  corresponding 
pairs  of  these  elements. 

Rema/rk.  By  using  the  system  of  oblique  projecting 
.  instead  of  the  usual  mode  of  perpendicular  ones, 
the  relative  positions  of  the  lines  projected  are  not  changed, 
since  these  lines,  being  all  parallel  to  the  assumed  vertical 
plane,  will  be  projected  on  it  in  their  true  lengths,  whether 
the  projecting  lines  be  oblique,  or  perpendicular  to  this 
plane.  By  the  system  of  perpendicular  projections,  a  sepa- 
rate construction,  like  the  first,  would  have  been  requisite  to 


STONE    CUTTIXG.  33 

determine  each  element ;  whereas  by  the  oblique  one  used 
the  construction  of  one  ellipse  DNn  and  of  the  line  BC"  are 
alone  sufficient. 

2d.  The  curve  of  right  section  must  lie  in  some  plane 
perpendicular  to  the  elements  of  the  oblique  cylinder.  To 
tix  such  a  plane,  draw  a  line  XY  perpendicular  to  the  pro 
jections  of  the  elements  on  the  horizontal  plane ;  and  from 
Y,  where  it  cuts  BD,  another  line  YZ  perpendicular  to  the 
projections  of  the  elements  on  the  assumed  vertical  plane. 
These  two  lines  may  be  taken  as  the  traces  of  a  plane  on 
these  two  planes ;  and,  as  these  traces  are  perpendicular  to 
the  projections  of  the  elements  on  the  two  planes,  the  plane 
itself  will  be  perpendicular  to  the  elements,  and  will  cut 
from  the  cylinder  a  right  section. 

To  construct  this  curve  of  right  section,  it  will  be  neces- 
sary to  find,  in  the  first  place,  on  the  assumed  vertical  plane, 
the  projections  of  the  points  in  which  the  elements  of  the 
oblique  semi  cylinder  pierce  the  plane  of  right  section.  To 
do  this,  it  is  evident  that  the  vertical  planes  which  contain 
these  elements,  as  the  one,  for  example,  of  which  CC\  and 
CO'  are  the  traces,  cut  the  plane  of  right  section  in  lines 
parallel  to  YZ,  its  trace  on  the  assumed  vertical  plane.  To 
find  the  projection  on  this  plane  of  the  line  cut  out  by  the 
vertical  plane  of  which  CC[  and  CO'  are  the  traces,  it  is 
plain  that  the  point  z,  where  the  horizontal  traces  XY  and 
CCl  intersect,  will  be  one  point  of  the  required  hue.  This 
point,  being  in  the  assumed  horizontal  plane,  will  be  pro- 
jected into  the  line  BX,  the  ground  line  of  the  assumed 
vertical  plane,  at  z',  by  drawing  a  line  through  z  parallel  to 
AB,  according  to  the  method  of  oblique  projections  adopted. 
If  from  z'  a  line  be  drawn  parallel  to  YZ,  this  line  z'z"  will 
be  the  required  projection  of  the  line  cut  from  the  plane  of 
right  section  by  the  vertical  plane  which  contains  the  ele- 
ment projected  in  GC.  The  point  z",  where  this  line  cuts 
C"C",  the  projection  of  this  element  on  the  assumed  verti- 
cal plane,  will  be  the  projection  on  this  plane  of  one  point 
of  the  curve  of  right  section.  In  like  manner,  projecting 
the  points  X,  x,  y,  &c,  into  the  ground  line  BX  at  X,  x\ 
y ,  &c,  and,  from  these  last  points,  drawing  parallels  to  YZ, 
the  points  X",  x",  y",  &c,  in  which  they  cut  the  correspond- 
ing elements  in  projection,  will  give  other  points ;  and  the 
curve  X'x"z"y"  Y  will  be  the  projection  of  the  curve  of 
right  section  required. 

Remark.  Since  the  elements  are  projected  on  the  as- 
sumed vertical  plane  into  their  true  lengths,  it  is  evident 
that  taking  any  point  of  this  projection  of  the  curve  of 


34  .  8T0XE   CUTTING* 

right  section,  as  x"  for  example,  the  distance  x"m"  will  be 
the  true  distance  of  the  point  of  which  x"  is  the  projection 
from  the  horizontal  semi  cylinder,  as  measured  on  the  ele- 
ment projected  in  n/"m,"  j  and  x"n'"  will  be  its  true  distance 
from  the  vertical  plane  containing  the  semicircle  A ' C' '  B '. 
In  like  manner,  the  true  distances  of  other  points  of  the 
curve  of  right  section  from  this  plane  and  from  the  horizon- 
tal semi  cylinder  measured  along  the  elements  may  be  found 
from  the  projection  of  this  curve. 

Having  found  the  projection  of  the  curve  of  right  sec- 
tion, the  curve  itself  can  be  found  by  revolving  the  plane 
of  right  section  upon  the  horizontal  plane  around  its  trace 
XY.  The  distance  zz^  of  the  point  projected  in  z"  from 
XY  \$>  evidently  equal  to  z'z",  since  this  line  is  the  pro- 
jection, in  its  true  length,  of  the  one  in  the  plane  of  right 
section  drawn  through  the  point  z  in  the  vertical  plane 
containing  the  element  projected  in  CG\  In  like  manner, 
the  distances  XX ^  xx„  yyn,  being  set  off  from  XY,  along 
the  perpendiculars  to  it  through  these  points,  and  equal 
respectively  to  the  distances  XX\  x'x",  &c,  the  curve 
X^z^y^  Yl  will  be  the  required  one.  The  line  Xt  Yl  which 
corresponds  to  X"  Y  in  projection  will  be  the  diameter  of 
this  curve. 

3d.  Having  the  curve  of  right  section  in  its  true  length, 
as  well  as  the  elements  of  the  oblique  cylinder,  and  the 
points  where  they  cut  this  curve,  it  will  be  easy  to  make 
the  required  developments  which,  for  convenience,  will  be 
done  on  the  assumed  vertical  plane  in  its  revolved  position. 
To  do  this  set  off  on  the  line  JTZ,  from  the  point  Y,  the 
distances  Y'z3,  Yxs,  YX^  respectively  equal  to  the  arcs 
Yxy%z„  Y&,  &c,  of  the  curve  of  right  section.  The  right 
line  YXt  will  be  the  development  of  this  curve.  Through 
the  points  z3,  x3,  &c,  thus  set  off,  draw  perpendiculars  to 
YX2,  these  will  be  the  indefinite  lengths  of  the  developed 
elements  drawn  through  the  points  z3,  x3,  &c.  From  these 
last  points  set  off  z3C3 ,  z3Ci  respectively  equal  to  z"C"  and 
z"C" j  in  like  manner,  set  off  the  distances  x3n3  and  x3m% 
respectively  equal  to  x"n"  and  x"m'",  &c.  The  curves 
B"C3n3An  and  DC4m3E/}  will  be  the  developments  respec- 
tively of  the  semicircle  A!G'B\  in  which  the  oblique  semi 
cylinder  cuts  the  given  vertical  plane  through  AB,  and  of 
the  curve  in  which  it  intersects  the  horizontal  semi  cylinder. 
The  developed  portion  of  the  oblique  semi  cylinder  which 
lies  between  these  curves  and  the  developed  positions  B"I) 
and  AXEX  of  its  bounding  elements  will  be  the  one  required, 


STONE   CUTTING.  35 

4th.  To  obtain  the  horizontal  projection  of  the  curve  in 
which  the  semi  cylinders  intersect.  Project  the  points  C" , 
m",  &c,  into  I?.zv,  by  perpendiculars ;  from  the  foot,  as  C„ 
of  each  perpendicular  draw  an  oblique  projecting  line 
parallel  to  AB;  the  points,  as  C[,  n„  ra„  &c,  in  which  these 
intersect  the  projections  of  the  corresponding  elements,  will 
be  points  in  the  required  projection ;  and  the  curve  DnxC;mxE 
will  be  the  one  required. 

Theorem  1,  {PI.  A,  Fig,  3.)  If  the  lines  AB  and  DE, 
which  bisect  each  other  and  are  horizontal,  are  the  transverse 
axes  of  two  semi  ellipses,  t/ie  planes  of  the  two  curves  being 
vertical,  and  having  the  same  semi  conjugate  axis, projected  in 
the  point  C,  then  will  the  lines  which  join  the  points  of  the 
curves  at  the  same  height  above  the  horizontal  plane  of  the 
transverse  axes  be  parallel  to  each  other. 

Let  A'C'B'  be  the  projection  of  the  semi  ellipse,  having 
AB  for  its  transverse  axis,  on  a  plane  parallel  to  itself;  and 
let  the  other  curve  be  revolved  about  the  common  semi 
conjugate  into  the  plane  of  the  first  and  be  projected  into 
D'U'E'.  Drawing  any  line,  as  m'n',  parallel  to  AB' ,  it 
will  cut  the  two  curves  at  the  points  ml,  n',  and  o',p',  at 
the  same  height  above  the  horizontal  plane ;  the  first  two 
being  projected  horizontally  in  m  and  n  ;  the  second  in  o, 
and  px  in  the  revolved  position  of  the  second  curve,  and  in 
o  and  p  in  its  original  position.  Joining  the  points  o  and 
m,  also  p  and  n,  then  will  these  lines  be  parallel  to  each 
other,  and  to  the  lines  AD  and  EB  which  join  the  lowest 
points  of  the  two  curves.  For,  from  the  properties  of  two 
ellipses  having  a  common  conjugate  axis,  the  corresponding 
ordinates  of  the  curves  to  this  axis  will  be  proportional  to 
their  semi  transverse  axes ;  that  is, 

en1  :cp'  ::  C"B'  :  C"E'-, 
or,  On:  Up  ::  CB  :  CE; 
by  substituting  the  equal  lines  in  horizontal  projection.  But 
when  this  last  proportion  obtains,  the  lines  pn  and  EB  are 
parallel.  In  like  manner,  mo  may  be  shown  to  be  parallel 
to  AB,  and  consequently  to  EB.  The  same  holds  true  for 
the  lines  mp  and  on,  with  respect  to  AE  and  DB.   • 

It  follows  from  this  that  two  semi  ellipses,  having  the 
above  conditions,  will  be  the  curves  of  intersection  of  two 
genii  cylinders,  the  axes  of  which  lie  in  the  horizontal  plane 
of  the  transverse  axes  of  the  curves,  and  are  parallel  respec- 
tively to  the  lines  joining  the  extremities  of  these  axes. 
The  converse  of  this  proposition  is  also  evidently  true,  viz. : 
if  the  axes  of  two  elliptical  or  circular  semi  cylinders  lie  in 
the  same  horizontal  plane  and  intersect,  and  the  highest  elo* 


36  STONE   CUTTING. 

ment  of  each  is  at  the  same  height  abo-\  t  this  plane,  then 
will  their  curves  of  intersection  be  plane  curves,  and  be  pro- 
jected on  the  horizontal  plane  in  the  two  right  lines  which 
join  the-  opposite  points  of  intersection  of  the  lowest  ele- 
ments. 

Theor.  2,  {PL  A.  Fig.  5.)  Let  A'CB'  be  the  vertical 
projection  of  a  send  ellipse  situated  in  a  vertical  plane,  of 
whtch  AD,  parallel  to  A7B',  is  the  horizontal  trace}  and  let 
tin  point  O,  on  theperp<  ndicular  OC  to  A15  which  bisects  it, 
he  the  horizontal  projection  of  a  vertical  Vine  through  O,  and 
let  the  semi  ellipse  and  this  vertical  be  taken  as  the  directrices 
of  a  surface,  generated  by  moving  a  line  parallel  to  the  hor- 
izontal  plane,  and  m  each  of  its  successi/vt  j  i  sitions  touching 
the  vertical  at  O  and  the  semi  ellipse y  then  will  any  section 
of  this  surface  by  a  plane,  as  ad,  parallel  to  the  plane  of 
the  ellipse-,  be  also  an  ellipse  of  wh/ich  the  line  ah  intercepted 
between  OA  and  OB,  will  be  one  axis,  and  the  line  OC  equal 
to  the  other  semi  axis.  And  a  tangent  line  drawn  to  this 
ellipse  at  any  point,  as  the  one  projected  in  n,  n',  urill  pierce 
the  horizontal  plane  in  a  line  OD,  draicnfrom  the  point  O 
to  thepoint  J),  where  a  tangent  to  the  directing  ellipse  at  the 
point  projected  in  m,  m',  at  the  same  height  as  n',  also  pierces 
the  horizontal  plane. 

1st.  Through  the  point  projected  in  n,  n' ,  draw  the  pro- 
jections On i.  and  o'm'  of  an  element  of  the  surface,  and 
project  the  line  ab  into  a'b '.  As  the  lines  BO  and  bo  are 
parallel,  there  obtains 

730 :  bo  ::  mO :  no; 

hut  BO=B'0';  bo=b'0',tmdmO=m'o';  no=n'o'; 
therefore,  BO'  :  I/O'  ::  m'o'  :  n'o', 
which  shows,  from  the  properties  of  ellipses  having  a  com- 
mon axis,  that  the  curve  b'n'C  is  an  ellipse. 

2d.  Let  tangents  be  drawn  to  the  two  ellipses  at  the  cor- 
responding points  m'  and  n',  at  the  same  height  above  A'B'. 
These  tangents  will  intersect  the  common  axis  at  the  same 
point  E' ,  and  the  other  axis  at  points  I)'  and  d'  such,  that 

01/  :  O'd'v.m'o  :  n'o'. 
Projecting  the  points  I)'  and  d'  into  the  respective  planes 
of  the  two  ellipses  at  D  and  d,  and  observing  that  OD'  = 
OB}  Od'=od/  m'o' =m ( ' ;  an.dn'o'=noj  there  obtains 

OB  :  Cm  ::  cd  :  <n; 
that  is,  the  line  joining  the  points  I>  and  d  passes  through 
the  point  0. 

Remark.  This  surface  is  a  right  conoid  of  which  the 
horizontal  plane  is  the  plane  director. 


BIOXE   CUTTING.  37 

Prok.  0.  To  construct  a  tangent  plane  and  a  normal, 
line  to  the  conoid  at  any  point,  as  n,  n'. 

Draw  the  projections  of  the  element  Om,  o'm!  of  the 
surface  at  the  given  point.  Find  where  the  tangent  to  the 
directing  ellipse  at  the  point  m,  m  pierces  the  horizontal 
plane.  Join  this  point  JJ  with  0.  Through  n  draw  a  par- 
allel to  CD,  and  where  it  cuts  OP,  at  d,  draw  a  line  XJ? 
parallel  to  Om.  This  is  the  horizontal  trace  of  the  tangent 
plane  at  n,  n' .  For  the  tangent  line  at  the  given  point  to 
the  ellipse  projected  in  ah  pierces  the  horizontal  plane  at  d, 
this  is  therefore  one  point  of  the  horizontal  trace  required, 
and  as  the  element  of  the  surface  is  contained  in  the  tangent 
plane  and  is  horizontal  the  trace  XYwill  be  parallel  to  Om.. 

The  line  zv,  drawn  through  n  perpendicular  to  XY,  is 
the  indefinite  projection  of  the  normal  to  the  surface  at 
n,  n'. 

Prob.  1,  {PI.  A,  Fig.  6.)  To  draw  a  tangent  plane  and 
a  normal  line  to  a  helicoidal  surface  at  a  given  point  on  the 
surface. 

Suppose  XYZ,  the  involute  of  any  given  curve  xyz,  to 
be  the  base  of  a  vertical  cylinder;  and  let  the  line  Zy' ,  tan- 
gent to  this  curve  at  Z,  be  the  horizontal  trace  of  a  plane 
tangent  to  the  cylinder  along  the  element  projected  in  Z; 
and  in  this  tangent  plane,  revolved  on  the  horizontal  plane, 
let  any  inclined  line  Zy"  be  drawn  through  the  point  Z.  If 
the  tangent  plane  be  now  returned  to  its  vertical  position, 
and  be  wrapped  around  the  vertical  cylinder,  the  inclined 
line  Zy"  will  form  a  helix  on  the  cylinder,  and  the  points 
y",  m",  &c,  in  their  position  on  the  cylinder,  will  be  pro- 
jected into  its  base,  at  the  points  Y,  m,  <fec. ;  such  that  the 
arcs  ZY,  Z/n,  will  be  equal  to  the  distances  Zy',  Zm' ,  &c, 
from  Z  to  the  projection  of  these  points  in  the  tangent 
plane  into  its  horizontal  trace.  If  a  right  line  be  moved 
along  this  helix  so  that  in  all  its  positions  it  shall  be  parallel 
to  the  horizontal  plane,  and  be  projected  on  this  plane  in 
lines  as  Zzt,  Yy,  normal  to  the  curve  XYZ,  this  line  will 
generate  a  helicoidal  surface,  the  elements  of  which  will  be 
normal  to  the  cylinder  of  which  XYZ  is  the  base,  and  tan- 
gent to  the  one  of  which  pyz1  is  the  base. 

Let  the  point  projected  in  Y,  and  which  is  at  the  height 
y'y"  above  the  horizontal  plane,  be  the  one  at  which  it  is 
required  to  construct  a  tangent  plane,  and  a  normal  line  to 
the  surface.  As  the  helix  makes  a  constant  angle,  equal  to 
the  one  Zy"y'',  with  the  elements  of  the  cylinder,  at  the 
points  where  they  intersect ;  and  as  a  tangent  to  the  helix 
at  any  point  n  akes  the  same  angle  as  the  helix  dues  at  this 


38  9T0XE   CUTTING. 

point  with  the  element  of  the  cylinder ;  it  ifl  evident  that 
the  tangent  to  the  helix,  at  the  point  projected  in  Y,  will 
pierce  the  horizontal  plane,  at  a  point  z,  on  the  tangent 
drawn  to  the  curve  at  Y,  at  the  same  distance  from  this 
point  as  the  distance  Zy'  —  ZY.  This  point  s  will  therefore 
be  one  point  of  the  horizontal  trace  of  the  required  tangent 
plane.  But,  as  the  tangent  plane  contains  the  horizontal 
element  of  the  surface  at  the  given  point,  its  trace  on  the 
horizontal  plane  will  he  parallel  to  this  element.  Drawing 
a  line  AB  through  z  parallel  to  Yy,  this  will  he  the  trace 
of  the  required  plane.  The  normal  will  evidently  he  pro- 
jected in  Yz,  as  this  line  is  perpendicular  to  AB  the  requir- 
ed trace. 

To  find  the  time  length  of  this  normal  let  CD,  parallel 
to  Yz,  he  the  trace  of  a  vertical  plane.  The  element  through 
the  given  point  will  pierce  this  plane  at  a  height  YY  above 
CD  equal  to  y'y ".  The  normal  line  will  be  projected  on 
this  plane  in  its  true  length,  and  will  pass  through  the  point 
Y.  But,  as  the  vertical  plane  through  CD  is  also  perpen- 
dicular to  the  tangent  plane  at  the  given  point,  the  line 
Yz'  will  be  the  projection  as  well  as  the  trace  of  the  tangent 
plane  on  this  vertical  plane.  The  line  YS  drawn  perpen- 
dicular to  Yz'  will  therefore  be  the  indefinite  projection  of 
the  required  normal. 

Remark.  The  traces  of  all  tangent  planes  to  the  surface, 
at  points  on  the  element  considered,  will  evidently  be  par- 
allel to  Yy,  the  projection  of  this  element;  and  the  trace 
of  any  one  may  be  readily  found  by  means  of  the  point  z. 
.For,  through  y,  where  the  projection  of  the  element  is  tan- 
gent to  the  curve  ocyz,  through  the  point  z  draw  an  indefinite 
line  yy^j  then  if  at  any  point  on  the  projection  of  the  ele- 
ment lya  line  be  drawn  parallel  to  Yz,  the  projection  of 
the  tangent  at  Y,  the  point  where  this  parallel  cuts  yyl  will 
be  a  point  in  the  horizontal  trace  of  the  tangent  plane  at 
the  assumed  point.  This  may  be  readily  proved  as  follows: 
Let  the  tangent  projected  in  Yz,  and  the  element  of  the 
cylinder  projected  in  y  be  taken  as  the  directrices  of  a  warp- 
ed surface  of  which  the  horizontal  plane  is  the  plane  director 
This  warped  surface  will  be  tangent  to  the  helicoidal  surface 
throughout  the  entire  element  projected  in  Yy;  for  they 
have  the  same  plane  director;  a  common  tangent  plane  at 
the  point  projected  in  Yj  and  also  one,  which  is  vertical, 
at  the  point  projected  in  y.  Now,  as  the  warped  surlace  in 
question  is  a  hyperbolic  paraboloid,  any  vertical  plane  par- 
allel to  Yz  will  intersect  it  in  a  right  line  which  will  cut  the 
element  projected  in  Yy,  and  pierce  the  horizontal  plane  ir 


STONE   CUTTING.  39 

the  line  yy,  wliere  the  trace  of  the  assumed  vertical  plane 
cuts  it.  This  point  will  therefore  be  a  point  in  the  trace  ol 
a  tangent  plane  to  both  surfaces  at  the  point  where  the  line 
cut  from  the  hyperbolic  paraboloid  intersects  the  element 
common  to  the  two  surfaces,  and  along  which  they  aro 
tangent. 

Prob.  7,  PI.  1.  To  construct  the  projections  and  true 
dimensions  of  the  bounding  lines  and  surfaces  of  the  vous 
soirs  of  a  horizontal  full  centre  arch. 

This  problem  comprises  several  cases,  according  to  the 
character  and  the  positions  of  the  surfaces  which  form  the 
ends  or  heads  of  the  arch. 

Case  1st.  The  arch  being  terminated  at  one  end  by  a  ver- 
tical plane  oblique  to  its  axis  and  at  the  other  by  a  vertical 
plane  perpendicular  to  the  axis. 

Let  the  semicircle  {Fig.  1)  of  which  B '  C  is  the  di- 
ameter, be  the  right  section  of  the  soffit  of  the  arch,  and 
having  divided  t it  into  any  odd  number  of  equal  parts,  as 
five  for  example,  draw  through  the  points  of  division  F\ 
F\  &c,  radii  which  prolong  to  the  semicircle  described  on 
A! D  as  a  diameter.  These  radii  will  be  directions  of  the 
joints  in  the  right  section,  A' '  B'  being  their  common  thick- 
ness. Through  the  points  A',  I\  G\  &c,  drawing  vertical 
and  horizontal  lines,  they  will  be  the  exterior  bounding  lines 
of  the  voussoirs  in  right  section ;  the  line  K'K"  which 
bounds  the  top  of  the  keystone  being  assumed  at  pleasure. 

Let  AD  {Fig.  2)  be  the  trace  on  the  horizontal  plane 
containing  the  axis  and  lowest  elements  of  the  arch  of  the 
vertical  plane  oblique  to  the  axis  which  forms  the  front  end 
of  the  arch,  and  ad  the  trace  of  the  one  perpendicular  to 
the  axis  that  bounds  the  back  end. 

As  the  edges  of  any  voussoir,  as  the  one  of  which  Q'M' 
N'O'P'  {Fig.  1)  is  the  right  section,  are  all  parallel  and 
horizontal,  they  will  be  projected  into  their  true  lengths  in 
plan  between  the  two  traces  of  the  end  planes.  The  one 
corresponding  to  M'  will  be  projected  in  plan  {Fig.  2)  in 
Mpnl ;  that  corresponding  to  N'  in  N1nl ,  &c. 

As  all  the  joints,  except  the  two  lowest,  are  oblique  to 
the  horizontal  plane,  their  horizontal  edges  alone  are  pro- 
jected into  their  true  dimensions  in  plan.  The  two  lowest, 
corresponding  to  A'B'  and  CD,  are  projected  into  AabB, 
and  CcdD  respectively.  To  find  the  true  dimensions  of  the 
others,  and  the  development  of  the  soffit,  develop  in  the  first 
place  the  right  section  of  the  soffit,  by  setting  oft'  the  dis- 
tances B'F",  E'F\  &c,  {Fig.  5)  respectively  equal  to 
the  arcs  B ' E' \  &c  {Fig.  1) ;   and  then,  by  Fig.  2,  PL  At 


40  8TONE    CUTTING. 

find  the  points  B,  Et,  Mn  &&,  of  the  developed  curve 
in  which  the  sollit  is  intersected  by  the  oblique  plane  of 
front  end;  also  the  right  line  a,  e, ,  mt ,  &c,  which  is 
the  development  of  the  right  section  in  which  it  is  cut  bv 
the  plane  of  the  back.  The  surface  bounded  by  the  devel- 
oped curves  and  the  two  lowest  elements  of  the  semi  cylin 
der  is  the  developed  soffit. 

To  find  any  oblique  joint  in  its  true  dimensions,  as  the 
one  projected  in  plan  between  the  lines  Mjm,  and  N ,n, ,  set 
off  from  M',  (1'i'j.  5)  to  the  right  along  ML"  a  distance 
equal  to  M 'N'  {Fig.  1),  the  breadth  of  the  joint.  From 
the  point  N'  {Fig.  5)  draw  a  line  perpendicular  to  JB  <  . 
and  from  N'  set  off  N'Nt  equal  to  n'NJ  {.Fig.  2).  Join  the 
points  M,NI ;  the  trapezoidal  figure  3f/]Y/n/m/  is  the  requir- 
ed joint.  In  like  manner  the  true  dimensions  of  all  the 
other  joints  are  found  as  shown  on  Fig.  5. 

Case  2d.  The  arch  being  terminated  at  one  end  by  a  plane 
oblique  to  its  axis  and  to  the  horizontal  plana  of  its  two  low- 
est elements  and  at  the  other  by  a  vertical  plane  perpendicu- 
lar to  the  axis. 

Let  AD  {Fig.  2)  be  the  horizontal  trace  of  the  oblique 
plane,  and  let  the  angle  which  it  makes  with  the  horizontal 
plane  be  f .  Drawing  any  line  vw  parallel  to  AD  {Prob.  1) 
as  the  projection  in  plan  of  a  horizontal  line  of  this  plane, 
and  from  any  point  as  A  a  perpendicular  Ax  to  vw,  the 
height  of  the  point  x  above  the  horizontal  plane  will  be 
three  times  the  distance  Ax.  Setting  off  from  A'  {Fig.  3) 
the  length  Ax  to  x',  at  a?' erecting  the  perpendicular  x'x"= 
ZA'x'  and  joining  A',  x",  the  angle  x'A'x"  will  be  the  angle 
between  the  oblique  and  horizontal  planes.  Taking  now 
the  distance  Ay  {Fig.  2)  in  which  the  line  vw  cuts  the  line 
Aa,  parallel  to  the  axis,  and  setting  it  off  from  A'  to  y' 
{Fig.  3),  erecting  a  perpendicular  y'y"=x'x",  and  joining 
A'y",  the  angle  yA'y"  will  be  the  one  between  the  oblique 
and  horizontal  ]  Janes  measured  in  the  vertical  plane  parallel 
to  the  axis  of  which  Aa  {Fig.  2)  is  the  trace. 

Having  drawn  the  line  A'y",  and  the  vertical  at  A', 
the  projections  in  plan  of  the  points  in  which  the  oblique 
terminating  plane  cuts  the  bounding  horizontal  lines  of  the 
voussoirs  corresponding  to  the  points  F',  1 ",  F',  &c.  {Fig.  1) 
in  right  section  are  readily  determined  by  applying  the  con- 
structions in  Prob.  4.  The  points  in  plan  corresponding  to 
E'  and  /'  {Fig.  1),  for  example,  Mill  be  found  by  drawing 
lines  through  E' ,  I  parallel  to  A'y',  taking  the  lengths 
E"E"\  A!' A'"  and  -dting  them  ofl  from  F)  and  /,  [Fig. 
2)  from  the  tra  :e  A]>  to  F  and  /,  along  the  projections  in 


STONE   CUTTING.  41 

plan  of  the  corresponding  edges  of  the  vcussoirs,  The 
curve  BEF,  &c,  thus  obtained,  and  the  pentagonal  figures 
EllIGF,  &c.j  are  the  projections  in  plan  of  the  olltique 
sections  of  the  soffit  and  voussoirs  of  the  arch. 

Remark.  As  a  verification  of  the  accuracy  of  the  con- 
structions, the  lines  IE,  GF,  &c,  prolonged  should  pass 
through  the  point  L  where  the  axis  cuts  the  trace  AD;  and 
the  lines  Atl,  JIG,  &c,  should  be  parallel  to  AD,  as  the 
top  surfaces  of  the  voussoirs  are  horizontal  planes. 

Case  3d.  The  arch  being  terminated  by  an  oblique  plane, 
as  in  either  of  the  preceding  cases,  and  at  the  other  extremity 
by  a  semi  circular  cylinder,  its  axis  and  two  bounding  ele- 
ments being  in  the  same  plane  as  the  corresponding  lines  of 
the  arch  and  perpendicular  to  them. 

Let  ad  {Fig.  2)  be  one  bounding  element  of  the  given 
semi  cylinder.  Having  set  off  the  radius  of  this  cylinder 
from  D,  {Fig.  4)  on  the  line  A'D  prolonged,  and  described 
a  portion  of  the  semicircle  tangent  to  the  vertical  DN" , 
draw  through  the  points  31,  W,  &c,  {Fig.  1)  parallel  lines 
to  A'D.  By  Prob.  4  find  the  projection  in  plan  of  bef,  &c, 
of  the  curve  of  intersection  of  the  soffit  of  the  arch  and  the 
semi  cylinder;  also  the  projections  of  the  points  i,  h,  g,  &c, 
in  which  the  horizontal  edges  of  the  voussoirs  intersect  the 
semi  cylinder;  the  pentagonal  figures  efghi,  &c,  will  be  the 
projections  in  plan  of  the  lines  in  which  the  surfaces  of  the 
voussoirs  intersect  the  semi  cylinder.  Any  point  in  plan,  as 
n,  is  found  by  setting  off  a  length  ntn  from  ad,  along  the 
projection  of  the  edge  corresponding  to  N' ,  equal  to  the  dis- 
tance N"N'"  {Fig.  4). 

Remark.  The  lines  ei,fg,  &c,  are  portions  of  ellipses, 
which  prolonged  pass  through  the  point  /,  in  which  the  axis 
of  the  arch  cuts  the  bounding  element  ad.  The  lines  gh, 
no,  &c,  are  right  lines,  being  the  projections  of  the  inter- 
section of  the  horizontal  surfaces  of  the  voussoirs  with  the 
semi  cylinder.  The  lines  hi,  op,  &c,  are  the  projections  of 
arcs  of  circles  in  which  the  side  vertical  planes  of  the  vous- 
soirs corresponding  to  H'l',  OP',  intersect  the  semi  cyl- 
inder. 

The  true  dimensions  of  the  joints  in  either  of  the  two 
last  cases  are  found  by  setting  off'  from  the  line  B'C  {Fig. 
5)  the  lengths  along  the  perpendiculars,  at  the  points  E',  I, 
&c,  which  correspond  to  the  distances  respectively  of  the 
points  E,  /and  e,  i,  {Case  Zd,  Fig.  2)  from  A'D. 

Remark.  As  a  verification  of  and  aid  to  accuracy  of 
construction,  let  a  line  LI  {Fig.  5)  be  drawn  parallel  to  the 
edge  M  'm  and  at  a  distance  fi  ^m  it  equal  to  I'M'  {Fig.  1) 


42  STOXE    CUTTING. 

the  radius  of  the  rig-lit  section;  the  right  lines  MJV  and 
M  Xt  prolonged  should  intersect  the  line  LI  at  the  same 
point  L,  such  that  the  length  LI  {Fig.  5)  shall  be  the  same 
as  IX  {Fig.  2).  In  like  manner  the  curve  .m/ m  prolonged 
should  intersect  the  same  line  LI  at  the  point  I  which  cor- 
responds to  the  one  I  {Fig.  2).  Similar  constructions  of 
verification  will  he  found  on  Fig.  5  {PU.  3  and  4). 

From  an  examination  of  Fig.  2  it  will  he  seen,  that  the 
projection  in  plan  of  the  voussoir  corresponding  to  M'JV'O', 
&G.,  {Fig.  1)  in  right  section,  will  in  Case  3d  be  hounded  on 
one  end  by  the  figure  MJTO,  &c.,  on  the  other  by  the  one 
mno.  &e. ;  and  by  the  parallel  hues  which  join  the  corres- 
ponding points  Mm,  ISn,  6zc. 

Application.  Having  found  the  principal  dimensions  of 
the  voussoirs,  let  it  be  required  to  cut  from  a  single  block  of 
stone  of  the  form  of  a  rectangular  parallelopipedon  the 
voussoir  corresponding  to  M'N'X)\  &c  {Fig.  1)  in  Case  3d. 
Drawing  a  line  S'T  through  Q'  perpendicular  to  0'P\  and 
prolonging  O'N'  to  R'  on  the  line  drawn  through  M'  par- 
allel to  0'P\  the  rectangle  R'T  will  evidently  be  the  di- 
mensions of  the  end  of  the  block  within  which  the  voussoir 
in  right  section  can  be  inscribed.  The  dimensions  of  the 
length  of  the  block  will  evidently  be  determined  by  drawing 
through  the  point  Q  {Fig.  2)  a  line  B ' Q  parallel  to  ro. 
Having  inscribed  on  the  end  of  a  block  of  the  form  and 
dimensions  thus  found,  the  figure  in  right  section,  the  block 
would  be  prepared  by  cutting  away  those  portions,  as 
M.S'  Q\  &c,  which  are  exterior  to  the  figure.  This  being 
done,  the  points  corresponding  to  M,  iT,  0,  &c,  and  m,  n, 
o,  &c,  {Fig.  2)  can  be  set  off  on  the  corresponding  edges, 
and  the  two  ends  of  the  voussoir,  the  one  terminated  by 
the  oblique  plane,  the  other  by  the  semi  cylinder,  be  obtain- 
ed. In  cutting  away  the  portions  of  the  block  to  form  the 
curved  surfaces  of  the  soffit  and  of  the  end  of  the  voussoir, 
a  model  cut  from  a  thin  board,  by  shaping  it  en  the  lack  to 
the  form  of  the  arc  M'Q'  {Fig.  1),  and  a  like  model  cut  to 
the  form  of  the  arc  DN"\  would  be  requisite  as  a  guide  to 
the  workman,  to  be  applied,  from  time  to  time,  in  a  direc- 
tion perpendicular  to  the  elements  of  the  cylinders,  until  it 
is  found  that  the  models  coincide  accurately  at  all  points 
with  the  prepared  surfaces. 

Models  also  of  the  true  forms  of  the  joints,  determined 
in  Fig.  5,  may  be  cut  from  thin  pasteboard,  or  any  like  ma- 
terial, and  be  used  to  verify  the  work.  These  last  would 
evidently  not  be  requisite  to  guide  the  workman  in  setting 
off  his  points  where  he  works  from  a  block  of  the  above 


STONE    CUTTING.  43 

form.  But,  in  cases  where  a  block  of  irregular  shape  has 
to  be  taken,  they  may  be  found  the  most  convenient  for  set- 
ting off  the  points  to  determine  the  form  of  these  joints  on 
the  stone. 

Prob.  8,  PI.  2.  To  construct  the  projections  and  true 
dimensions  of  the  voussoirs  in  the  groined  and  cloistered 
arches. 

In  each  of  these  cases  the  soffits  of  the  arches  are  formed 
by  the  intersections  of  two  semi  cylinders,  the  axes  of  which 
are  in  the  same  horizontal  plane,  and  their  top  elements  at 
the  same  height  above  this  plane.  From  these  conditions, 
the  curves  of  intersection  of  the  soffits  (Theor.  1)  will  be 
plane  curves,  and  will  be  projected  in  plan  in  the  diagonal 
lines  which  join  the  intersections  of  the  lowest  elements  of 
the  semi  cylinders. 

In  the  cases  selected  to  illustrate  this  problem  (PI.  2), 
the  curve  of  right  section  of  one  of  the  semi  cylinders  is  a 
semicircle  {Fig.  1),  that  of  the  other  (Fig.  2)  a  semi  ellipse, 
each  having  the  same  rise  L  K '•  and  their  axes  are  taken 
perpendicular  to  each  other.  The  joints  in  each  arch  cor- 
responding to  F'G',  &c,  are  normal  to  the  soffit,  or  surfaces 
of  their  respective  cylinders ;  the  upper  and  lower  edge3  of 
the  corresponding  joints  in  each  arch  being  in  the  same 
horizontal  plane,  as  well  as  the  top  surfaces,  as  G'H'  (Figs. 
1,  2),  of  the  voussoirs. 

Remark.  Fig.  1  is  the  right  section  of  the  semi  circular 
arch ;  Fig.  2  that  of  the  semi  elliptical  arch ;  Fig.  3  above 
the  line  ah  is  a  portion  of  the  plan  of  the  groined  arch,  the 
soffit  of  the  serai  circular  arch  being  projected  within  the 
angle  BKG  and  the  corresponding  soffit  of  the  semi  ellipti- 
cal arch,  only  the  half  of  which  is  shown  in  plan,  being  pro- 
jected within  the  angles  aKB  and  bKC.  Fig.  5  on  the 
right  represents  two  of  the  joints  of  a  groin  stone  belonging 
to  the  semi  circular  arch  with  the  development  of  the  por- 
tion of  the  soffit  between  them ;  that  on  the  left  the  true 
dimensions  of  the  corresponding  parts  of  the  same  stone 
which  forms  a  part  of  the  other  arch. 

Fig.  4  below  the  line  ab  is  a  portion  of  the  plan  of  the 
cloistered  arch,  the  soffit  of  the  semi  elliptical  portion  being 
projected  within  the  angles  BIKGI :  that  of  the  semi  circu- 
lar portion  being  projected  within  the  angles  B2KC/  and 
B/F02 .  Fig.  6  on  the  right  represents  the  two  joints  of  a 
groin  stone  which  forms  a  part  of  the  semi  circular  arch 
with  the  portion  of  the  soffit  between  them ;  that  on  the  left 
the  corresponding  parts  of  the  same  stone  of  the  other  arch, 


44:  STONE   CUTTING. 

Groined  Arch.  Having  constructed  (Fig.  1)  the  right 
section  of  the  semi  circular  arch  as  in  Prob.  7,  assume  Ji  (  " 
{Fig.  2)  as  the  transverse  axis  of  tlie  ellipse  of  right  section 
of  the  other  arch,  and  placing  it  at  any  convenient  position, 
on  the  left,  perpendicular  to  the  direction  B'(  "  (Fig.  1),  set 
off  the  semi  conjugate  L'K'  equal  to  the  radius  of  the  semi- 
circle, and  describe  the  semi  ellipse.  Find  on  the  semi 
ellipse,  as  shown  by  the  lines  A'  />  \  &c,  on  the  left  of  Fig. 
1,  the  points  J.\  /•',  &c.5  at  the  same  height  above  B'C,  as 
the  corresponding  points  of  the  semicircle.  ( ionstruct  tan- 
gents to  the  semi  ellipse  at  these  points,  and  at  the  same 
normals  for  the  directions  of  the  joints.  Find  on  these 
normals  the  points  T,  G',  etc.,  at  the  same  heights  as  the 
like  points  in  Fig.  1.  Through  the  points  /',  6r,  (Fig.  2) 
draw  the  vertical  and  horizontal  lines  I'll'  and  G'H',  for 
the  bounding  lines  of  the  voussoirs.  Having  completed,  in 
this  way,  the  right  section  (Fig.  2),  draw,  in  plan  (Fig.  3), 
the  projection  of  the  axes,  and  the  bottom  elements  of  the 
arches,  corresponding  to  the  points  B\  C  (Figs.  1,  2). 
Drawing  the  diagonal  lines  BB,  and  CC\  between  the  points 
where  these  elements  intersect  in  plan,  they  will  be  (Theor. 
1)  the  projections  of  the  ellipses  in  which  the  two  semi  cyl- 
inders intersect,  and  which  form  the  edges  of  the  groins. 

To  find  the  projections  in  plan  of  the  voussoir  of  the  groin 
which  corresponds  to  the  one  M'N'O',  &c.  (Fig.  1),  and 
F'G'R',  &c.  (Fig.  2),  draw  Mm  (Fig.  3),  the  projection  of 
the  lower  edge  of  the  joint  corresponding  to  31'  (Fig.  1), 
and  Jlmi  the  corresponding  projection  for  the  point  F'  (Fig. 
2) ;  and  in  like  manner  the  lines  Nn  and  Nnt ,  the  projec- 
tions of  the  upper  edges.  Joining  M  and  N  gives  the  pro- 
jection of  the  intersection  of  the  planes  of  the  two  joints. 
Find  in  like  manner  Qq  and  Qqt  •  Pjp  and  1'jr  ,  the  projec- 
tions of  the  edges  corresponding  to  the  joints  Q'P'  and 
EI'.  Joining  Q  and  P  gives  the  projection  of  the  inter- 
section of  the  planes  of  the  two  last  joints.  Having  found 
the  projections  in  plan  of  the  edges  of  the  joints  and  their 
intersections,  the  voussoir  is  terminated  on  the  semi  circular 
arch  by  a  joint  of  right  section  mp,  taken  at  any  suitable 
distance  from  the  point  P  j  and  on  the  semi  elliptical  arch 
by  a  like  joint  iMJ>r  The  required  voussoir  in  plan  will  be 
the  figure  Mrn/pPjp^iht .  The  part  above  the  line  MG  be- 
longing to  the  semi  circular  arch ;  that  to  the  right  of  it  to 
the  other. 

To  find  the  true  dimensions  of  the  joints  of  this  voussoir 
and  of  the  portion  of  thesofnt  which  belongs  to  the  semi  <  ir- 
cular  arch,  draw  a  line  mp  (Fig.  5)  to  correspond  to  the  one 


STONE    CUTTING.  4:5 

mp  {Fig.  3)  of  the  plane  of  right  section  by  which  the  vous- 
soir  is  terminated.  On  this  line  set  off  mn=M'JV'  {Fig.  1) 
the  breadth  of  the  upper  joint;  nq=M'Q'  the  length  of 
the  arc  between  the  joints;  and  qp=QP'  the  breadth  of 
the  lower  joint.  Through  these  points  draw  perpendiculara 
to  the  line,  and  set  off  on  them  nN=mM  {Fig.  3) ;  mM= 
nN ;qQ=qQ,  &n&pP=pP.  Join  the  points  31,  J\T,  and 
Q,  P,  by  right  lines ;  the  points  JV,  Q,  by  a  curve  line,  an 
intermediate  point  of  which  can  be  found  by  constructing 
in  plan  the  element  yx  of  the  soffit  which  corresponds  to 
the  middle  point  x'  {Fig.  1)  of  the  arc  3FQ'  and  setting 
this  line  off  on  Fig.  5  from  y,  the  middle  point  of  nq  to  x.  The 
Fig.  5  will  give  the  joints  required  in  their  true  dimensions, 
also  the  developed  portion  of  the  soffit  between  them,  and 
which  is  bounded  at  one  end  by  the  curve  of  right  section 
corresponding  to  mq,  and  at  the  other  by  the  portion  of  the 
ellipse  of  the  groin  corresponding  to  MQ  {Fig.  3). 

In  like  manner  the  true  dimensions  of  the  joints,  and 
the  portion  of  the  soffit  between  them,  which  belong  to  the 
same  stone,  and  form  the  portion  of  the  elliptical  arch  ter- 
minated by  the  joint  of  right  section  mipi  {Fig.  3)  may  be 
found,  as  shown  {Fig.  5)  on  the  left ;  by  setting  off  the  dis- 
tances p  q, ,  q/ml ,  mlnl ,  respectively  equal  to  E'l',  F'F' 
and  F'G'  {Fig.  2),  and  on  the  perpendiculars  to  plnl ,  through 
the  points  j?,  ,  qt ,  &c,  setting  off  the  distances  p,P,  q,Q,  &c, 
respectively  equal  to  pP  &c,  in  plan  {Fig.  3). 

The  portion  of  the  keystone  which  forms  the  top  of 
the  groins  at  the  point  K  {Fig.  3),  is  limited  on  the  semi 
circular  arch  by  the  joint  of  right  section  GtJVt  j  and  by  the 
one  Hh  on  the  semi  elliptical  cylinder,  with  a  corresponding 
one  on  the  right  of  K. 

The  joints  of  right  section  of  the  different  courses,  as 
Trip  and  GNt ,  are  arranged  to  break  joint. 

Cloistered  Arch.  The  constructions  for  determining 
the  projections,  &c,  of  the  joints  and  their  true  dimensions, 
are  precisely  the  same  in  this  case  as  in  the  preceding. 

In  Fig.  4c  aapj)  are  the  exterior  lines  in  plan,  and 
BfiBfi^  the  interior  lines  of  the  top  of  the  walls,  or  the 
imposts  of  the  arches ;  the  semi  circular  arch  springing  from 
the  lines  BnG  and  GB,  and  the  semi  elliptical  form  GtBt. 
The  lines  GtK  and  BtK  qxq  the  projections  of  the  half  of 
each  groin. 

By  drawing  in  plan  the  edges  of  the  joints  corresponding 
to  P'Q'  and  M'JY'  {Figs.  1,  2),  and  joining  the  points  P, 
Q,  and  3f,  N  {Fig.  4),  the  intersection  of  these  joints  are 
obtained  in  plan.     The  groin  stone  which  corresponds  to 


46  8T0XE   CUTTING. 

these  joints  is  limited  by  a  plane  of  right  section  nip  (Fig.  4) 
taken  at  pleasure  on  the  semi  circular  arch,  and  a  like  <>ne 
///  ji  on  the  semi  elliptical.  All  the  parts  of  this  groin  stone 
-will  therefore  be  projected  within  the  figure  iwpPv  m  JJ '. 

To  construct  the  joints  and  the  portion  of  the  soffit  in 
their  true  dimensions  which  belong  to  this  stone,  commence 
{Fig.  6)  by  setting  off  on  a  right  line  the  distances  nm,  ma, 
and  qp,  respectively  equal  to  S'J/',  21  Q ,  and  QP'  (Ftg. 
1),  and  which  correspond  t-o  the  joint  of  right  section  mm 
{Fig.  4).  Draw  through  the  points  n,  m,  &c,  thus  set  oft-, 
perpendiculars  to  np,  and  along  these  perpendiculars  set  off 
the  distances  nN,  m2L.  qQ,  and  pP,  respectively  equal  to 
the  same  lines  on  Fig.  4.  Join  the  points  NM  and  QP  by 
right  lines;  and  21 Q  by  a  curved  line,  an  intermediate  point, 
of  which  corresponding  to  x  {Fig.  1)  is  found  by  setting  off 
from  yt  the  middle  of  mq  the  distance  yixi  equal  yxt  {Fig.  4). 

In  the  same  way  the  corresponding  portions  of  the  groin 
stone  belonging  to  the  joint  of  riijht  section  m/p/  on  the 
semi  elliptical  arch  are  found  from  Figs.  2  and  4. 

The  top  groin  stone  at  K  {Fig.  4)  which  forms  a  portion 
of  the  two  arches  is  represented  as  a  single  stone. 

The  joints  of  right  section  in  the  different  courses  of 
voussoirs  are  arranged  as  shown  in  plan  to  break  joint. 

Application.  Having  determined  the  projections  in  plan 
of  the  edges  of  the  joints  of  a  groin  stone  with  the  true 
dimensions  of  the  joints,  and  the  portion  of  the  soffit  of 
each  arch  belonging  to  it,  their  uses  in  shaping  the  stone 
from,  the  solid  block  will  be  easily  understood.  Taking,  for 
example,  the  groin  stone  of  the  groined  arch,  the  right  sec- 
tions of  which  are  given  in  Figs.  1  and  2;  the  plan  in  Fig. 
3;  and  the  true  dimensions  of  the  joints,  &c,  in  Fig.  5,  it 
will  be  readily  seen  that,  supposing  a  block  from  which  the 
stone  is  to  be  shaped  to  be  a  rectangular  parallelopid,  its 
thickness  must  be  such  that  the  right  section  21'JV'O',  &c, 
{Fig.  1)  can  be  inscribed  within  the  rectangle  of  the  end 
that  corresponds  to  the  joint  of  right  section  vp  {Fig.  3)  of 
the  semi  circular  arch,  and  the  figure  F'G'JB.  ,  Arc,  be  in- 
scribed within  the  end  corresponding  to  the  joint  m/pl  of 
the  semi  elliptical  arch;  and  the  length  of  the  block  must 
be  equal  to  rnM,  and  its  breadth  to  Mm  . 

•Having  inscribed  upon  the  ends  of  the  block  the  two 
figures  of  cross  section,  the  portions  of  the  solid  exterior  to 
them  are  gradually  worked  off  until  the  dressed  surfaces 
coincide  with  Fig.  5,  which  may  be  ascertained  either  by 
measurement,  or  by  the  actual  application  of  these  figured 
cut  from  some  thin  flexible  material. 


STONE   CUTTING.  47 

Remark.  A  careful  examination  of  the  lines  of  tho 
figures  will  show  the  geometrical  methods  for  determining 
the  tangents  to  the  points  on  Fig.  2  which  correspond  to 
Fig.  1. 

■  Prob.  9,  Pis.  3,  4.     To  construct  tJte  projections  and  true 
dimensions  of  the  voussoirs  of  the  rampant  cylindrical  arch. 

This  problem,  which  comprises  two  cases,  is  a  variation 
of  Prob.  7.  The  arch  in  this,  as  in  Prob.  7,  being  termi- 
nated at  one  end  by  a  vertical  plane,  and  at  the  other  by  a 
horizontal  semi  cylinder  having  its  axis  and  elements  paral- 
lel to  the  vertical  plane  of  the  end ;  the  elements  of  the  soffit 
and  the  edges  of  the  voussoirs  of  the  arch  being  oblique  both 
to  the  vertical  plane  and  to  the  horizontal  plane  of  the  plan. 

Case  1,  PL  3.  Construct  (Fig.  1)  the  semicircle  B'T'C, 
to  represent  the  oblique  section  of  the  arch  by  the  vertical 
plane  of  the  end.  Let  ad  (Fig.  2),  assumed  at  any  conve- 
nient distance  from  A'D',  be  the  projection  in  plan  of  the 
lowest  element  of  the  semi  cylinder  of  the  other  end,  which 
with  the  axis  is  taken  in  a  horizontal  plane  at  the  distance 
a" 'a"  (Fig.  3)  below  the  line  A'D' ;  and  let  the  lowest  ele- 
ments of  the  arch  drawn  from  the  points  A'D' CD',  and  its 
axis  from  F ,  (Fig.  1)  be  taken  to  intersect  the  line  ad,  and 
to  lie  in  vertical  planes  perpendicular  both  to  the  vertical 
plane  of  the  end  and  to  the  horizontal  plane.  The  edges 
of  the  voussoirs  as  A'a,  B'b,  (fee,  in  plan  will  be  perpendic- 
ular to  A'D'. 

To  obtain  the  edges  of  the  voussoirs  in  their  true  dimen 
sions,  it  will  be  necessary  to  find  their  projections,  as  in  Prob. 
5,  on  a  plane  parallel  to  them. 

Let  the  vertical  plane  which  contains  the  edge  projected 
in  A'a  be  taken  for  this  purpose,  and  suppose  it  revolved 
around  the  line  A '  K'1' ,  its  trace  on  the  vertical  plane  of  the 
end,  to  coincide  with  this  plane.  In  this  new  position  of 
tl>e  side  vertical  plane  the  edge  projected  in  A  a  will  be  ob- 
tained on  it  by  setting  off  A'a'"=A'a  /  erecting  at  a'"  a 
perpendicular  =  a'"a",  and  joining  A'a'".  The  edges 
drawn  from  B',  C ,  D',  and  the  axis  of  the  arch  will  all 
evidently  be  projected  into  the  same  line  A'a". 

The  projections  of  the  other  edges  on  this  plane  will 
evidently  be  parallel  to  A'a'  (Fig.  3),  and  their  positions 
will  be  found  by  drawing  through  the  points  E',  I',  <fec, 
(Fig.  1)  lines  parallel  to  A'D',  and  from  the  points  E  ",  &c, 
in  which  they  cut  A'K'"  drawing  parallels  to  A'a"  (Fig.  3). 

As  a'  is  the  revolved  position  of  the  point  in  which  the 
vertical  side  plane  cuts  the  lowest  element  of  the  semi  cyl- 
inder of  the  end,  and  as  the  axis  is  in  the  same  horizontal 


48  STOKE    CUTTING. 

plane  as  this  element,  by  setting  off  on  the  horizontal  through 
a",  and  to  the  left  of  a",  the  radius  of  the  semi  cylinder 
and  from  the  point  thus  set  off  describing  an  &rca"X',  it 
will  be  the  one  cut  from  the  semi  cylinder  by  the  vertical 
side  plane.  The  lines,  as  H"II'",  E"E '"  intercepted  between 
the  arc  a"X'  and  the  line  AX",  are  the  projections  in  their 
true  dimensions  of  the  required  edges  of  the  voussoirs. 

To  obtain  a  right  section  of  the  arch,  for  the  purpose  of 
constructing  the  joints  of  the  voussoirs  in  their  true  dimen 
sions,  and  the  development  of  the  soffit  of  the  arch ;  from  the 
point  A'  draw  the  line  A'  Y'  perpendicular  to  the  projec- 
tions of  the  edges  on  the  vertical  side  plane ;  this  line  may 
be  regarded  as  the  trace  of  a  plane  of  right  section  on  the 
side  vertical  plane,  and  the  line  A'D'  as  its  trace  on  the 
horizontal  plane  through  A'D'.  If  the  plane  of  right  sec- 
tion thus  fixed  be  revolved  around  A'D'  until  it  becomes 
horizontal,  the  right  section  contained  in  it  will  be  deter- 
mined in  its  true  dimensions.  The  points  as  <?',  f,  &c, 
{Fig.  4)  will  be  fouud,  after  this  revolution,  by  setting  oft' 
from  the  line  A'D'  along  the  projections  of  the  elements  in 
plan  corresponding  to  the  points  E ',  F',  &c,  distances  equal 
to  A'e',  A'f',  &c,  {Fig.  3)  measured  from  the  point  A'  along 
the  line  A'  Y'.  The  curve  B'e'f'm'q'C  {Fig.  4)  thus  deter- 
mined is  the  curve  of  right  section  of  the  soffit,  and  the  fig- 
ure 7n'n'o'j?'q'  the  right  section  of  the  voussoir  corresponding 
to  M'X'O'P'Q'  {Fig.  1)  of  the  end. 

Having  obtained  the  right  section,  the  development  of 
the  soffit  and  the  joints  in  their  true  dimensions  are  found, 
as  in  Prdbs.  5  and  T,  as  follows :  Havin^drawn  a  line  B'  C 
{Fig.  5)  set  off  along  it  the  distances  B'e\  e'f',f'm',  &c, 
respectively  equal  to  the  arcs  Be',  e'f  &c,  on  the  curve  of 
right  section  {Fig.  4).  The  right  line  B'C  will  be  the  de- 
velopment of  the  curve.  Through  the  points  B',  e',f,  &c. 
draw  perpendiculars  to  B'C  which  will  be  the  elements  of 
the  soffit  in  development,  which  are  the  lower  edges  of  the 
joints  and  correspond  to  the  points  B\  E\  &c,  {Fig.  1), 
As  the  true  distances  of  the  extremities  of  these  edges  from 
the  plane  of  right  section  are  given  on  Fig.  3,  and  are  the 
distances  e'e",  e'E'"  for  the  edge  projected  in  its  true  length 
between  the  line  A'E'"  and  the  curve  d'X'j  by  setting  off 
e'e"  and  e'E'",  from  e'  to  E',  and  e'  to  E'"  {Fig.  5),  the 
points  E'  and  E'"  will  be  respectively  points  of  the  devel- 
opment of  the  curves  in  which  the  soffit  intersects  the  verti- 
cal plane  of  the  one  end  of  the  arch  and  the  horizontal 
c}Tlinder  that  terminates  the  other.  In  like  manner  the 
points  F',  M\  &c,  of  the  developed   curve  B'T'C  are 


8T0NE   CUTTING.  49 

found,  and  those  as  h",  E'",  &c,  of  the  other  end.  To  ob- 
tain the  true  dimensions  of  any  joint,  as  the  one  correspond- 
ing to  EI'  {Fig.  1),  set  off  the  distance  e'i',  {Fig.  5)  equal 
to  the  breadth  of  the  joint  in  right  section,  which  is  e'i1 
{Fig.  4).  Through  i'  draw  a  perpendicular  to  B'C,  and  set 
off  from  it  the  distance  i'l',  i'A'",  respectively  equal  to 
a' A",  a' A'"  {Fig.  3).  Join  I',  E' ,  by  a  right  line,  and 
A"E'"  by  a  curved  line,  the  figure  A'"I'E'E'"  will  be  the 
required  joint  in  true  dimensions.  The  other  joints  are 
found  in  like  manner. 

To  find  the  dimensions  of  a  block  of  the  form  of  a  rec- 
tangular parallelopiped  from  which  one  of  the  voussoirs,  as 
the  one  corresponding  to  M'N' 0' P' Q'  could  be  cut,  it  will 
be  observed  that  the  edges  of  this  voussoir  are  projected  in 
their  true  dimensions  on  Fig.  3,  between  the  line  H"  H'" , 
which  corresponds  to  the  points  N' ,  0'  {Fig.  1),  and  the  line 
E"E'",  which  corresponds  to  Q';  drawing  therefore  from 
H"  a  perpendicular  to  E"E'"  prolonged,  and  from  E"  one 
to  H" Hf"  prolonged,  the  rectangle  thus  formed  will  be  the 
true  dimensions  of  one  side  of  the  block.  As  m'n'o'p'q' 
{Fig.  4)  is  the  cross  section  of  the  same  voussoir,  the  breadth 
of  the  rectangle  of  the  end  of  the  block  must  be  equal  to 
m'p\  in  order  that  the  figure  m'n'o' ,  &c,  can  be  inscribed 
within  it.  The  manner  of  setting  off  the  different  lines  on 
the  block,  with  the  view  of  dressing  it  into  shape,  will  be 
readily  seen  from  what  has  already  been  stated  on  this  point 
in  the  preceding  problems. 

Remarks.  From  the  preceding  constructions,  the  joints 
and  the  development  of  the  soffit  for  any  other  plane  end 
passing  through  the  line  A  '  D' ,  and  having  the  line  A'Z',  for 
example,  for  its  trace  on  the  vertical  side  plane,  can  be 
readily  found,  by  setting  off,  on  Fig.  5,  from  the  line  B'C, 
the  distances  between  the  corresponding  points  on  the  lines 
A'  Y'  and  A'Z',  as  h'h",  for  example,  in  the  same  way  as 
for  those  between  A'  Y'  and  A '  K" ' ,  and  through  the  points 
e",f",  &c,  drawing  the  developed  curve  of  intersection  of 
the  soffit  and  assumed  plane,  and  constructing  the  corres- 
ponding joints  as  e"i"A'"E"'. 

Prdb.  9,  Case  2d,  PI.  4.  This  case  is  a  variation  of  the 
preceding  one,  the  axis  and  elements  of  the  soffit  of  the 
arch  being  oblique  both  to  the  horizontal  plane,  and  to  the 
vertical  plane  which  terminates  the  arch  at  one  end,  but 
situated  in  vertical  planes  oblique  to  the  vertical  plane  of 
the  end.  The  position  of  the  semi  cylinder  which  terminates 
the  other  end  is  the  same  in  all  respects  as  in  the  preceding 
case. 


50  STONE   CUTTING. 

Fig.  1  represents  the  end  of  the  arch  in  the  vertical 
plane.  The  curve  of  the  soffit  B'F'M '  C  in  this  plane  is  a 
semicircle.  The  arch  for  the  mere  illustration  of  this  case 
consists  of  only  three  voussoirs. 

Fig.  2  represents  the  projections  of  the  elements  in  plan  ; 
the  axis  of  the  arch  and  the  lowest  elements  of  its  soffit 
intersect  the  lowest  element  of  the  semi  cylinder,  which  is 
projected  in  the  line  ad,  parallel  to  AD',  and  lies  in  a  hor- 
izontal plane  at  the  distance  JD'D"  below  AD',  at  the 
points  b,  I  and  c. 

Fig.  3  represents  the  projections  of  the  edges  of  the 
voussoirs  on  the  vertical  side  plane,  parallel  to  them,  of 
which  D'T  and  D'  V  are  the  traces  on  the  horizontal  plane 
of  the  plan,  and  the  vertical  plane  of  the  end.  The  system 
of  projecting  lines  in  this  case  is  the  same  as  the  one  usee 
in  Prob.  5. 

Fig.  4  represents  the  revolved  position,  on  the  horizontal 
plane,  of  the  right  section  of  the  arch,  contained  in  the  plane 
of  right  section  of  which  XY,  perpendicular  to  the  projec- 
tion of  the  axis  L'l  is  the  horizontal  trace,  and  TV,  perpen- 
dicular to  D"d,  the  projection  of  the  axis  on  the  side  vertical 
plane,  is  the  trace  on  this  plane. 

Fig.  5  represents  the  joints  and  the  development  of  the 
soffit  in  their  true  dimensions. 

Having  constructed  Figs.  1  and  2,  find,  by  Prob.  5,  the 
projections  of  the  edges  of  the  joints  on  the  vertical  side 
plane  of  which  D'T  is,  the  horizontal  trace,  assuming,  in  the 
first  place,  the  line  D"d,  as  the  projection  of  the  axis  and 
lowest  elements  on  this  plane,  and  parallel  to  which  all  the 
other  projections  of  the  edges  are  drawn ;  the  one  corres- 
ponding to  the  point  M'  (Fig.  1),  for  example,  is  found  by 
setting  off  from  D"  {Fig.  3)  on  the  revolved  position 
D'K"'  of  the  trace  of  the  side  vertical  plane  with  that  of 
the  end,  the  distance  D'M"  equal  to  MM'  {Fig.  1),  and 
drawing  M"M'"  parallel  to  D"d. 

Representing,  by  the  line  drawn  through  T  parallel  to 
ad,  the  axis  of  the  semi  cylinder,  the  ellipse  cut  from  the 
semi  cylinder  of  the  end  will  have  dT,  and  TS'  equal  to 
the  radius  of  the  semi  cylinder,  for  its  semi  axes.  Having 
described  the  quadrant  dS'  of  the  ellipse  in  its  revolved  po- 
sition, the  projections  of  the  edges  of  the  joints  intercepted 
between  it  and  the  line  D'K'"  will  be  the  true  lengths  of 
the  edges. 

To  obtain  the  right  section,  take  XY  perpendicular  to 
the  axis  L'l  in  plan,  and  Yz'  perpendicular  to  its  projection 
ty'd  on  the  vertical  side  plane,  as  the  traces  of  the  plane 


STONE   CUTTING.  51 

of  right  section.  Having  found  {Fig.  4,  PI.  A,)  the  projec- 
tions c\  mt ,  n',  &c.  {Fig.  3)  of  the  points  in  which  the  ele- 
ments of  the  soffit  and  the  edges  of  the  joints  pierce  the 
plane  of  right  section,  next  construct  from  these  {Fig.  4)  the 
right  section  as  revolved  on  the  horizontal  plane. 

To  construct  the  soffit  and  joints  in  their  true  dimensions, 
draw  {Fig.  5)  a  line  atD'  {Fig.  4,  PL  A)  and  set  off  on  this 
from  b,  to  ct  the  length  of  the  curve  of  right  section  b.c/ 
{Fig.  4).  Drawing  through  the  points  b/ ,  jt  ,  ra, ,  c, ,  per- 
pendiculars to  afi' ,  set  off  on  them  above  and  below  atD' 
distances  m  M'  m  if'",  respectively  equal  to  m  3f"  and 
m,M '"  {Fig.  3).  The  curves  B'F'M. '  C,  and  bM'hc,  drawn 
through  the  points  thus  obtained,  will  be  the  developments 
of  the  intersection  of  the  soffit  with  the  end  plane  and  semi 
cylinder.  To  construct  the  joint  of  which  M'M'"  is  one 
edge,  set  off  m/n/  on  a.E'  equal  to  mlnl  {Fig.  4),  the  width 
of  the  joint  in  cross  section ;  through  n/  draw  a  parallel  to 
M' IF",  and  set  off  on  it  ntN' ,  n^tf"' ',  respectively  equal 
to  n'N"  and  n'JV'"  {Fig.  3).  Joining  M' N'  by  a  right 
line,  and  M'"W"  by  a  curve  line,  the  figure  obtained  is  the 
required  joint.     The  others  are  found  in  like  manner. 

Remark.  A  comparison  of  the  lines  on  Fig.  5  with 
those  on  Fig.  3  will  point  out  the  manner  of  constructing 
an  intermediate  point  as  v  of  the  curve  M" ' N'" . 

Prob.  10,  PI.  5.  To  construct  the  projections  and  true 
dimensions  of  the  voussoirs  of  the  hemispherical  dome. 

Let  the  semicircle  {Fig.  1)  BL"C  be  the  vertical  section 
of  the  soffit  of  the  dome,  and  suppose  it  divided  into  seven 
equal  parts  at  the  points  E\  Fr,  &c.  Drawing  radii  through 
the  points  F',  &c,  set  off  upon  them  the  equal  distances 
E'l',  F'G',  &c,  and  complete,  as  in  the  preceding  cases, 
the  figures  JEP'1I"G'F',  &c,  to  represent  the  sections  of 
the  voussoirs.  If  Fig.  1  be  supposed  to  be  revolved  about 
the  vertical  radius  PL"  as  an  axis,  any  section  of  a  vous- 
soir,  as  EIH"G'Fr  would  generate  the  entire  voussoir  of 
the  dome  comprised  between  the  horizontal  circles  on  the 
6offit  projected  in  the  lines  E'Q'  and  F'M'.  The  lines 
EI',  F'G',  in  this  revolution  generate  the  joints  between 
the  voussoir  in  question  and  the  two  in  contact  with  it, 
which  joints  are  portions  of  a  cone  of  which  the  centre  of 
the  dome  is  the  vertex,  and  LL'  the  axis.  The  line  F El' 
will  generate  a  cylindrical  surface  having  the  same  axis,  and 
the  line  H'G'  a  plane. 

Having  in  this  manner  determined  the  bounding  surfaces 
of  each  course  of  voussoirs,  the  course  is  divided  into  blocks 
of  suitable  dimensions,  by  joints  of  right  section,  formed  by 


52  STONE   (TUTTING. 

intersecting  the  course  by  vertical  planes  through  the  axis 
L'L".  If  IE,  and  IIFI  {Fig.  2),  for  example,  be  taken  aa 
the  projections  in  plan  of  two  joints  of  right  section,  the 
figure  ll  F ,E \  will  be  the  projection  in  plan  of  a  block  01 
voussoir  of  the  course  in  question.  The  figure  E'I'I'F" 
{Fig.  1)  is  the  projection  of  the  lower  conical  joint  of  this 
voussoir;  F  G'G"F"  that  of  the  upper  conical  joint;  and 
E'FF"E"  that  of  the  portion  of  the  soffit.  The  joints 
of  right  section  of  the  adjacent  courses  break  joints,  aa 
shown  at  E'E',  V"v',  &c,  on  the  curves  V'L",  Y"L', 
&c.  {Fig.  1),  which  are  the  projections  of  the  circles  cut 
from  the  soffit  by  the  joints  of  right  section. 

Application.  Having  determined  the  projections  of  the 
bounding  lines  and  surfaces  of  a  voussoir,  their  true  dimen- 
sions can  be  easily  determined,  and  from  them  the  size  of  a 
block  from  which  the  voussoir  can  be  cut.  Taking,  for  ex- 
ample, the  voussoir  projected  in  plan  {Fig.  2)  in  II \F \Et , 
from  an  inspection  of  the  projections  {Figs.  1,  2)  it  is  obvious 
that  the  dimensions  ,of  a  block  from  which  it  can  be  cut 
must  be  such  that  the  figure  IIFIEI  {Fig.  2)  can  be  inscribed 
within  its  base,  and  its  thickness  be  equal  to  the  vertical 
height  between  the  lines  H'G"  and  E'E"  {Fig.  1),  the  tot  a" 
depth  of  the  voussoir.  Having  selected  a  block  of  the  suit- 
able dimensions,  the  different  lines  and  surfaces  of  the  vous- 
soir can  be  obtained  in  their  true  dimensions  from  its  projec- 
tions {Figs.  1,  2,  3),  and  marked  out  on  the  sides  and  bases 
of  the  block. 

It  will  be  seen  that  the  end  of  this  voussoir,  which  forms 
a  portion  of  the  soffit,  is  comprised  between  the  two  merid- 
ian planes  IE,  and  I/F/  {Fig.  2)  and  the  upper  and  lower 
conical  joints.  The  points  F',  F",  E\  E",  {Fig.  1)  are 
therefore  the  projections  of  the  four  angular  points  of  the 
voussoir,  on  the  soffit,  and  lie  upon  the  circumference  of  a 
small  circle  of  the  dome  passing  through  the  points  of  which 
these  are  the  projections  on  Fig.  1.  This  small  circle  can 
be  readily  constructed  {Fig.  4),  since  the  lengths  of  the 
chords  joining  the  two  upper  points  FF",  and  the  two 
lower  E'E"  {Fig.  1),  are  given  in  their  true  dimensions  EFt 
and  EF  {Fig.  2);  and  the  diagonals  projected  in  EFt  and 
E,F{Fig.  2)  can  be  readily  obtained  in  their  true  dimensions. 
Having  set  out  the  small  circle  determined  from  these  ele- 
ments {Fig.  4),  it  will  limit  the  portion  of  the  soffit  on  the 
end  of  the  voussoir,  and  will  serve  as  a  guide  to  the  work- 
man in  working  it  out. 

Fig.  3  gives  the  true  dimensions  of  the  side  of  the  vous- 
soir  in  the  meridian  plane  IE,  {Fig.  2). 


STONE   CUTTING.  53 

Prdb.  11,  PI  6.  To  construct  the  projections  and  trite 
Ibnensions  of  the  voussoirs  of  the  gate-recess. 

The  line  BG  {Fig.  2)  represents  the  trace  of  the  vertical 
face  or  front  of  a  wall ;  ad  that  of  the  back,  also  vertical. 
Through  this  wall  an  arched  gate-way  is  to  be  so  constructed, 
that  the  gate,  composed  of  two  leaves,  may  be  placed  mid- 
way between  the  face  and  back,  as  at  AD,  and  when  open 
the  leaves  shall  be  thrown  back,  taking  respectively  the 
positions  Ata,  D,d,  by  revolving  around  the  vertical  axes 
projected  in  A,  and  D/ .  In  this  way,  the  gate  occupies  a 
recess  within  the  wall,  from  which  circumstance  the  problem 
is  named. 

Let  B',  F',  0',  (Pig.  1)  be  the  curve  of  right  section  of 
the  right  arch,  BBI C,  G  (Fig.  2)  its  plan.  Let  the  top  of 
the  gate  when  closed  be  a  semicircular  cylinder,  AD'  (Fig. 
1)  being  its  diameter,  and  the  rectangle  A  A  D  D  (Fig.  2) 
its  plan ;  the  gate  when  closed  shutting  against  the  plane 
surface  ring  projected  (Fig.  1)  between  the  two  semicircles; 
and  Fig.  2  in  the  line  AD. 

The  problem  to  be  solved  consists  in  so  arranging  the 
surface  of  the  recess  under  which  the  leaves  swing  in  being 
opened  or  closed :  1st,  that  it  shall  offer  no  obstruction  to 
the  play  of  the  leaves ;  2d,  that  it  shall  be  one  of  easy  geo- 
metrical construction ;  3d,  that  it  shall  present  a  pleasing 
architectural  effect. 

The  lines  A,a,  D,d,  (Fig.  2)  being  the  traces  of  the  ver- 
tical side  planes  of  the  recess  against  which  the  leaves  rest 
when  closed,  these  planes  are  each  terminated  at  top  by  an 
arc  of  a  circle  assumed  at  pleasure,  but  of  greater  radius 
than  A'L'  (Fig.  1).  To  construct  this  arbitrary  arc  (Fig.  3), 
revolve  the  side  plane  Dtd  around  the  axis  projected  in  D, 
(Fig.  2),  and  D'D'  (Fig.  3),  parallel  to  the  face  of  the  wall, 
into  the  position  D/d/ .  Assume  dl"  the  highest  point  of 
the  arbitrary  arc  in  the  revolved  position,  at  the  same  height 
as  Til  (Fig.  1)  is  above  X',  and  construct  an  arc  passing 
through  D'dl"  and  tangent  to  D'D  ,  and  let  this  be  taken 
for  the  required  arc.  Supposing  the  side  plane  revolved 
back  to  its  primitive  position,  Del''  will  be  the  projection 
of  the  arc ;  and  d'd"  that  of  the  vertical  edge  of  the  back 
and  side  planes. 

Let  this  arbitrary  arc,  the  semicircle  projected  in  A'lc  'D 
(Fig.  1)  and  AtDt  (Fig.  2),  and  the  axis  of  the  arch  pro- 
jected in  L'  (Fig.  1),  LI  (Fig.  2),  be  taken  as  three  direc- 
trices of  a  warped  surface,  to  form  a  portion  of  the  upper 
surface  3f  the  recess.     The  projections  of  the  extreme  posi- 


54  STONE   CUTTING. 

ticii  of  the  element  of  this  surface  will  be  d"x'L'  {Fig.  1\ 
day  {Fig.  2).     A  like  surface  covers  the  opposite  side. 

To  form  the  top,  the  two  warped  surfaces  determined  are 
connected  by  a  third,  which  must  be  tangent  to  each  of  them 
along  the  extreme  element  d"L',  a"L'  of  each,  so  that  the 
three  surfaces  may  appear  as  a  continuous  surface,  and  thus 
satisfy  the  3d  condition.  To  satisfy  this  condition,  let 
the  axis  of  the  arch  and  the  semicircle,  which  are  two  of 
the  directrices  of  the  two  first  warped  surfaces,  be  taken 
as  two  of  the  directrices  of  the  third.  This  will  give  two 
tangent  planes  common  to  the  surfaces  along  each  of  the 
elements  d'F,  a"L '.  Construct  now  a  tangent  plane  to 
the  warped  surface  found  at  the  point  d",  by  drawing  a  tan- 
gent to  the  curve  projected  in  D'd"  at  this  point,  and  through 
this  tangent  and  the  element  projected  in  d"  L  passing  a 
plane.  .  The  element  pierces  the  vertical  plane  of  which 
A  Dt  is  the  trace  at  x,  a?'/  the  tangent  to  the  curve  D \d"  at 
d"  intersects  the  vertical  line  D'D"  at  D";  joining  then 
D  ,  x',  it  will  be  the  projection  of  the  trace  of  the  tangent 
plane  on  the  vertical  plane  AD,  j  the  projection  of  its  trace 
on  the  vertical  plane  ad  is  v'd"w',  parallel  to  x'D".  Draw- 
ing an  arc  of  a  circle  passing  through  a"  and  d"  and  tangent 
to  v'w',  if  it  is  taken  as  the  third  directrix  of  the  second 
surface,  the  two  surfaces  will  be  tangent,  as  they  have  a 
third  common  tangent  plane  at  d". 

The  2d  condition  Is  satisfied  by  taking  warped  surfaces 
to  form  the  soffit,  or  top  surface. 

Having  constructed  the  warped  surfaces,  with  these  arbi- 
trary conditions,  it  will  be  necessary  to  ascertain  whether  thej 
satisfy  the  1st  condition.  To  do  this,  it  will  be  observed 
that  the  top  of  the  leaf  describes,  in  its  revolution,  a  surface, 
and  which,  to  satisfy  this  condition,  should  not  intersect  the 
warped  surfaces  within  the  side  plane,  as  Aa,  for  example. 
This  intersection  will  be  found  by  the  usual  methods  for 
finding  the  intersections  of  two  given  surfaces.  The  line 
r's'  {Fig.  1),  for  example,  may  be  assumed  as  the  vertical 
trace  of  a  horizontal  plane  intersecting  the  two  surf; 
This  plane  will  cut  from  the  surface  described  by  the  top  of 
the  leaf  an  arc  of  a  circle,  projected  in  sr  {Fig.  2)  and  from 
the  warped  surfaces  an  arc  8%trj  and  as  these  intersect  at  ?', 
without  Alta,  the  surfaces  do  not  interfere  along  this  hori- 
zontal plane.  The  same  construction  would  be  made  for 
other  points. 

The  hounding  lines  of  the  top  surface  being  found,  the 
arch  is  divided  off  into  five  equal  parts,  as  B'A  ,  ivc.  The 
planes  of  the  voussoir  joints  pass  through  the  axis  of  the 


STONE   CUTTING.  55 

arch  and  extend  to  the  points  /',  67',  &c,  arbitrarily  chosen 
and  from  these  last  points  the  voussoir  joints  are  vertical. 

To  determine  the  true  dimensions  of  the  joints  M'X 
and  Q  P  {Fig.  1);  let  each  of  them  be  revolved  around 
their  lower  horizontal  edges,  projected  in  M',  Q  ',  parallel  to 
the  horizontal  plane;  taking  JUf'JST',  this  is  done  by  draw- 
ing through  M '  a  line  parallel  to  L'D'  and  setting  off  along 
it,  from  21'.  distances  equal  to  G'F'=M'N\  F'i',  F  h', 
and,  to  simplify  the  construction,  drawing  through  the 
points  thus  set  off,  lines  parallel  to  L'M'  to  intersect  L'D'; 
from  these  last  points  drawing  lines  parallel  to  L'l  the  points 
LfoF'i'h'G'I  {Fig.  4)  will  be  found,  which  joined  will  give 
the  figure  and  true  dimensions  of  the  joint  through  M'N'. 
In  a  like  manner  the  figure  Fk[ E 'if  'g '  G '  1 and  true  dimen- 
sions of  the  joint  through  Q  P'  are  obtained. 

Application.  To  cut  the  voussoir  out  of  a  block  of  the 
form  of  a  rectangular  parallelopiped,  its  dimensions  must 
be  such  that  the  figure  N'jI'QP'O'  {Fig.  1)  can  be  inscrib- 
ed within  the  end,  and  its  length  be  equal  to  IG'  {Fig.  4). 
Having  set  out  the  bounding  lines  of  the  different  surfaces 
from  Fig.  1,  the  plane  and  cylindrical  portions  will  be  first 
cut  off;  next  the  portion  of  the  warped  sarfaces,  by  first 
working  down  to  the  positions  of  several  of  the  intermediate 
elements,  determined  from  the  drawing,  and  then  finishing 
off  by  the  eye  the  portions  of  the  surface  between  these  ele- 
ments. 

Prob.  12,  PI.  1.  To  construct  the  projections  and  true 
dimensions  of  the  steps  of  the  geometrical  stairioay. 

Let  ABGD  {Fig.  1)  be  the  polygonal  base  of  a  vertical 
wall,  along  which  a  flight  of  stone  steps  is  to  be  built.  Let 
XYZ,  xyz,  be  two  curves  having  the  relation  of  involute 
and  evolute  to  each  other ;  the  one  XYZ  being  the  base  of 
a  vertical  cylinder,  the  surface  of  which  limits  the  ends  of 
the  steps,  and  which  is  termed  the  well  of  the  stairs.  Let 
X \  Y,Z /  be  another  curve  parallel  to  the  one  XYZ,  and  at 
the  distance  from  it  that  persons  going  up  or  down  the  stairs 
would  naturally  take;  and  where,  on  this  account,  the  top 
of  each  step,  or  the  tread  should  have  a  uniform  breadth. 
This  tread  added  to  the  height,  or  rise,  being  usually  assumed 
at  twenty-two  inches,  as  a  convenient  distance  for  each  step. 

The  problem  consists :  1st,  in  arranging  the  tread  and 
rise  with  these  conditions ;  2d,  in  making  the  under  sur- 
face of  the  stairway  a  helicoidal  surface ;  3d,  in  arranging 
the  joints  between  the  steps  so  as  to  be  plane  surfaces,  and 
normal  to  the  helicoidal  surface  at  the  middle  p<  nt;  4th,  in 


56  STOXE    CUTTING. 

determining  from  these  conditions  the  form  and  dimensions 
of  each  step. 

Having  set  off  the  equal  arcs  XI,  1-2,  2-3,  &c.  (Fig. 
1)  along  the  curve  Xtl  tZt,  equal  to  the  assumed  tread, 
draw,  through  these  points,  lines  tangent  to  the  curve  an/z, 
and  prolong  them  to  the  line  bed  parallel  to  BCD /  the 
quadrilaterals  thus  formed,  between  X.  YZ  and  bed,  will  he 
the  true  dimensions  of  the  top  surface  of  each  step.  ]\i  id- 
way  between  the  equal  arc,  as  at  Y, ,  Z l}  cvjc,  draw  lines 
also  tangent  to  soys,  and  let  these  be  assumed  as  the  projec- 
tions of  "the  edges  of  the  joint3  along  the  helicoidal  surface ; 
and,  to  fix  their  position,  let  the  edge  of  each  joint  be  taken 
at  the  distance  of  half  the  rise  below  the  top  of  the  step. 
The  points  thus  determined  will  lie  on  a  helix,  which  at 
each  of  these  points  is  half  a  rise  below  the  top  of  the  step, 
and  the  inclination  of  the  tangent  to  which  at  any  point 
will  be  the  rise  or  height  of  each  step  divided  by  the  uniform 
tread. 

Having  fixed  the  position  of  the  helix,  the  helicoidal 
surface  is  generated  by  moving  a  right  line  along  it  so  as  to 
be  parallel  to  the  horizontal  plane,  and,  in  all  of  its  posi 
tions,  be  projected  normal  to  the  curve  XYZ. 

Let  yt ,  the  middle  point  of  the  lower  edge  Ym,  be  taken 
as  the  point  at  which  a  normal  plane  is  to  be  passed  to  the 
helicoidal  surface  for  the  joint  in  question.  This  plane  is 
determined  as  shown  on  Fig.  2,  by  Prob.  7  {PI.  A,  Fig.  6), 
and  in  like  manner  the  one  at  zt  on  Zo  as  shown  in  Fig.  3. 

Having  determined  these  planes,  their  intersections  with 
the  tops  of  the  steps  will  give  the  lines  Xn,  Op,  parallel  to 
Ym,  Zo,  which  are  the  top  edges  of  the  joints. 

"With  the  data  now  determined,  the  form  and  dimensions 
of  the  ends  of  the  step  to  which  these  two  joints  belong 
can  be  determined.  The  larger  end  of  the  step  is  contained 
in  a  vertical  plane  of  which  mq  is  the  trace  on  the  horizontal 
plane.  Draw  a  line  B  C  parallel  to  be  and  at  any  assumed 
distance  from  it;  this  may  be  taken  as  the  revolved  position 
of  the  top  line  of  the  ena,  on  the  horizontal  plane.  From 
the  points  m,  n,  o,  p,  and  q  draw  perpendiculars  to  mq', 
set  off  on  these  the  distance  q"q'  for  the  rise  of  the  step ; 
m"m'  equal  to  half  a  rise;  o'  half  a  rise  below p' ;  join  the 
points  thus  set  off.  The  figure  m'n'q"qp'o'  is  the  one  requir- 
ed. To  find  that  of  the  other  end  (Fig.  5),  the  portion  of  the 
cylinder  of  the  well  YM  is  developed,  and  the  correspond 
ing  points  set  off  on  it;  the  figure  Y'N'M"2£'0'Z'  is  the 
we  requi-ed. 


STONE   CUTTING.  57 

Application.  To  cut  the  stone,  a  block  must  be  taken 
upon  the  top  of  which  the  figure  YmqM  {Fig.  1)  can  be 
set  off"  and  on  the  large  end  Fig.  4.  Having  dressed  off 
the  plane  and  cylindrical  surfaces,  the  portion  of  the 
warped  surface  can  be  dressed  off  as  in  the  preceding  prob- 
lem. 

Prdb.  12,  PI.  8.  To  construct  the  projections  and  true 
dimensions  of  the  voussoirs  of  the  groined  annular  and 
radiant  arch. 

The  soffit  of  this  arch  is  formed  of  the  surfaces  of  an  an- 
nular and  radiant  arch,  the  intersections  of  which  form  the 
curves  of  the  groin. 

To  find  the  horizontal  projections  of  these  curves,  let  (Fig. 
2,  PI.  8)  B'  C'  be  the  diameter,  and  ~K"  the  centre  of  a  semi- 
circle, contained  in  a  vertical  plane  passed  through  the  ver- 
tical line  projected  at  L  horizontally.  If  the  semicircle  be 
revolved  around  the  vertical  L  it  will  generate  the  surface  of 
an  annular  arch. 

From  the  point  L  (Fig.  1)  in  the  horizontal  plane  of  the 
diameter  B'  0'  of  the  semicircle  let  two  lines  L  B'  and  L  c 
be  drawn,  making  any  convenient  angle  with  each  other,  and 
let  the  chord  of  the  arc  B'  c,  included  between  them,  de- 
scribed by  the  point  B'  of  the  semicircle,  in  its  revolution, 
be  taken  as  the  transverse  axis  of  a  semi-ellipse,  contained  in  a 
vertical  plane  of  which  B'  c  is  the  trace;  the  conjugate  semi- 
axis  of  this  ellipse  being  equal  to  the  radius  of  the  semicircle. 
Then  if  a  right  line  be  so  moved  that  it  shall,  in  all  of  its 
positions,  be  parallel  to  the  horizontal  plane  of  the  semi- 
transverse  axis  of  the  ellipse,  intersect  the  vertical  through 
L,  and  rest  on  the  curve  of  the  semi-ellipse,  it  will  generate 
the  soffit  of  the  radiant  arch. 

The  horizontal  projections  B'  L,  C,  and  C  L  c,  which  are 
those  of  the  groins,  will  be  obtained,  by  finding  the  inter- 
sections of  the  projections  of  the  corresponding  elements  cut 
from  the  two  soffits  by  horizontal  planes.  The  point  L, 
being  that  of  the  highest  point;  and  the  points  B;  C  Cj  c 
being  those  of  the  lowest  points  of  the  groins. 

If  now  the  semicircle  be  divided  into  five  equal  parts,  and 
the  right  sections  of  the  voussoirs  of  a  cylindrical  arch  be 
drawn  (Fig.  2) ;  the  joints  E'  I',  F'  G',  &c,  of  this  right  sec- 
tion, in  the  revolution  of  the  vertical  plane  containing  them, 
will  describe  zones  of  conical  surfaces,  the  vertices  of  which 
will  be  on  the  vertical  through  L,  where  these  joints  pro- 
longed intersect  it.  In  like  manner,  the  lines  I'  H',  G'  G", 
&c,  will  describe  cylindrical  surfaces,  having  the  same  verti- 
cal for  their  common  axis;  and  the  horizontal  line6,  as  G'  H', 


58  STONE   CUTTING. 

&c,  will  describe  zones  of  circles.  Thus  completing  the 
bounding  surfaces  of  the  voussoirs  belonging  to  the  annular 
arch. 

As  the  soffit  of  the  radiant  arch  is  a  right  conoid,  having 
the  horizontal  plane,  containing  the  semitransverse  axis  of 
the  directing  semi-ellipse,  for  its  plane  director,  and  the  ver- 
ticil through  L  as  its  right  line  directrix,  its  joints,  along  the 
rectilinear  elements  corresponding  to  the  horizontal  circles 
described  by  the  points  Q',  M',  F',  E',  of  the  semicircle,  to 
be  normal  throughout  to  the  soffit,  would  require  to  be 
warped  surfaces.  To  avoid  the  inconvenience  of  constructing 
these,  a  plane  surface  joint  is  substituted  for  each  stone  in- 
stead of  the  other;  and  this  is  so  taken,  that  it  shall  be  nor- 
mal to  the  soffit  at  the  middle  point  of  that  portion  of  the 
right  line  element  of  the  soffit  which  belongs  to  the  joint. 
Taking  for  example  the  element  projected  in  L  E,  e4,  and  as- 
suming that  the  groin  voussoir  is  bounded,  on  the  radiant 
portion  of  the  arch,  by  the  lines  F,/,;  by/,  I,  which  is  the 
projection  of  the  line  cut  from  the  soffit  by  a  vertical  cylin- 
der ;  by  the  projection  F,  E,  of  the  groin  curve,  and  by  the 
line  E'  ft,;  then  F,  /  will  be  the  lower  edge  of  the  radiant 
joint,  corresponding  to  the  lower  edge  of  the  annular  joint, 
described  by  the  point  F';  and  E,  <?2  will  be  the  lower  edge 
of  the  joint  below,  corresponding  to  the  one  E'  I'. 

Having  the  lower  edge  E,  es  of  the  joint,  the  other  bound- 
ing lines  of  it  are  found  by  constructing  a  normal  plane  to 
the  soffit  containing  the  right  line  element  through  E„  at  the 
middle  point  e  of  E,  e„  and  finding  the  intersections  gs  I  of 
this  plane  with  the  vertical  cylinder  f  I  that  limits  the 
voussoir;  with  the  conical  zone  described  by  E'  F,  and  which 
is  projected  in  E,  z  ;  and  finally  with  the  horizontal  plane 
which  passes  through  F  (Fig.  2),  and  which  will  be  projected 
through  I  parallel  to  E,  ev 

In  like  manner  the  plane  joint  projected  in  F,  f  gx  Gr  can 
be  found. 

The  portion  of  the  groin  stone,  belonging  to  the  annular 
arch,  is  limited  by  a  vertical  plane  passed  through  the  points 
L,  F„,  H;  the  line  F,  E,  of  the  groin;  the  upper  and  lower 
conical  joints  ;  the  cylindrical  surface  projected  in  z  H  ;  aud 
the  horizontal  plane"th rough  G'  H'  (Fig.  2). 

Note. —  To  construct  the  normal  planes  of  the  plane  joints 
of  the  radiant  arch  see  Prob.  6. 

All  the  horizontal  lines  of  the  surfaces  bounding  the  groin 
voussoir  in  question,  are  projected  in  their  true  dimensions 
n  Fig.  1. 

Having  found  the  horizontal  projections  of  all  the  lines 


STONE   CUTTING.  59 

bounding  the  groin  stone  considered,  the  true  dimen- 
sions of  all  the  developable  surfaces  by  which  it  is  bounded 
can  be  found  by  methods  already  used  in  the  preceding 
Probs. 

Take,  for  example,  the  plane  joint  of  the  radiant  arch, 
projected  in  E,  <?2  I  z.  Having  first  determined  the  tangent 
plane  to  the  middle  point  e%  of  the  lower  edge  of  the  joint, 
and  its  trace  t,  t,  on  the  horizontal  plane  of  the  springing 
lines,  by  Prob.  6,  let  this  last  plane  be  intersected  by  a  ver- 
tical plane  X  Y  perpendicular  to  the  lower  edge  E  e2  (Fig.  1), 
and  let  it  then  be  transferred,  parallel  to  itself,  to  X  Y  (Fig. 
3).  The  plane  X  Y  will  cut  from  the  tangent  plane  a  line 
which,  in  the  revolved  position  of  the  plane  X  Y  (Fig.  3), 
will  be  projected  in  t,  E' ;  E"  E'  (Fig.  3)  being  equal  to  E" 
E'  (Fig.  2).  From  E'  (Fig.  3)  drawing  a  line  perpendicular 
to  E'  4  it  will  be  a  normal  to  the  soffit  of  the  radiant  arch  at 
e2  (Fig.  1),  and  where  this  normal  intersects  at  I'  a  line  par- 
allel to  E''  t„  and  at  the  same  height  above  it  as  I'  (Fig.  2) 
is  above  E'^the  line  E'  I'  (Fig.  3)  will  be  the  true  width  of 
the  plane  joint  considered.  Now,  revolving  this  plane  joint 
around  the  line  E'  I'  to  coincide  with  the  vertical  plane  X  Y 
(Fig.  1),  the  points  ev  E:  (Fig.  1),  for  example,  will  fall  in  a 
perpendicular  to  E'  I'  (Fig.  3)  as  far  from  it,  at  e"  and  E", 
as  they  are  in  horizontal  projection  from  X  Y  (Fig.  1).  In 
like  manner,  the  points  i'  and.  1'"  (Fig.  3)  are  found ;  and 
e"  E"  V"  i  will  be  the  true  dimensions  of  the  plane  joint. 
E"  V"  wall  be  the  intersection  of  the  plane  radiant  joint  with 
the  corresponding  conical  joint  of  the  annular  arch  ;  and  e' 
i'  the  intersection  of  the  same  joint  with  the  cylindrical  joint 
fx  I  of  the  radiant  arch. 

Fig.  4,  showing  the  true  dimensions  of  the  upper  plane 
joint,  is  constructed  by  a  like  series  of  operations. 

Fig.  5  is  the  development  of  the  cylindrical  joint  of  the 
radiant  arch  projected  in/",  I;  and  of  the  cylindrical  surface 
of  the  groin  stone  of  the  annular  arch  projected  in  z  H. 

The  projections  of  the  conical  joints  of  the  annular  arch 
are  easily  found,  by  developing  the  cones  to  which  they  be- 
long. 

Fig.  6  is  the  development  of  the  end  surface  of  the  exterior 
voussnir  of  the  radiant  arch  which  joins  the  groin  voussoir 
considered. 

These  arches  rest,  as  in  the  cylindrical  groined  arch,  on 
pillars.  The  tops  of  these  pillars,  on  a  level  with  the  spring- 
ing lines  of  the  arches,  are  shown  in  the  trapezoids  B  B'  a'  a, 
CC'b'  b,  &c.  (Fig.  1.) 

The  dimensions  of  the  block  for  the  groin  stone  in  ques- 


60 


STONE   CUTTING. 


tion  area1'  given  in  the  projections,  sections,  &c,  of  Figs.  1, 
2,  3,  4,  6,  and  the  developments  of  the  conical  joints.  With 
„hese  elements,  the  bounding  lines  can  be  marked  out  on  the 
block,  and  the  voussoir  be  worked  off,  by  methods  similar  to 
those  pointed  out  in  the  two  preceding  problems. 


'\3  \TO,  >"i 


O.Rmnfft. 


Fiff.l 


r 


r 


Pl.l. 


O  JtoLnSK 


PI. 2 


H*fi%4if 


1-H — ,^_j — i    J-Nffi     ;    i 


♦itimm/  nm»ninm.nj-:  -  * 


7lfM!imn,7"'?.''!"ffi'K ...3  p 


llll«ilHIIIIII>i/ri.iiiiiiiii.i,iin.i,,r.,..ii,.,1 


0 RanM 


V. 


■•--.. 

\ 

\  \\ 

\ 
\ 

i      1     |C'  !    D'i 

1 

r 


PI.3. 


O.Hanm, 


l.>. 


Fiji. 


Fig.  5. 


~~"T~^» 

C'            D 

r             a 

O.San 


r 


2T;   [li  .-fj'S\       ^-'     V 


^i^JjXL 


---J.Bt-T** 


CLTiJfc. 


*    <■_ 


$ 

e 

! 

^ 

/i . 

_J 

J? !-- 

ft] 
u 

7H~" 

"7 - r* 

f*> 

t 

K    «J  > 4' 


>"' 


i  i  -AUr 


,^i 


<T 


H 


11" 


O.K^rtiTr 


1 

a  I    «/  i/ij;  v/jf  '  Ul 


r 


O.  RaTiftt. 


PI. 


V 


! 


/   / 


O.Rartfft. 


Pl.O. 


9 


eL^.-***- 


'-. 


Ik 


•* u 


ih    Vfc 


1     /;/ 

imi'iiiiiiiiil  ^t  /   / 


"^^Nio,       /[*         - 


II 


'// 


A-M'iL-..  i£z# — z 


.    ~-~ /      ■x.y;        afe--.    Br*-.-  ■       -x     \    I  • 

•5 ::; >/>-/?^^^  A  / 


<&■= 


/     !  \  \ 


i 

! 


PI -p. 


kU^'%3- 


■ 


University  of  California 

SOUTHERN  REGIONAL  LIBRARY ^LITY 

405  Hilgard  Avenue,  Los  Angeles ;CA90024  1388 

Return  this  material  to  the  library 

from  which  it  was  borrowed. 


~T 


■*•'*  I 


IT 


Date:     Thu,  19  Sep  91  12:47  PDT 

To        ECL4BAT 

Subject:  SRLF  PAGING  REQUEST 


Deliver  to 
Shelving  # 


UCSD  CENTRAL 

A   000  085  639  3 


Item  Information 

Mahan,  D.  H.   (Dennis  Hart^ 
Descriptive  geometry*  as  a 

Item 

ORION    #       :     2817y*t?7MC 


1802-187 
lied  to 


Requester  Inform 
Unit  :  UNK 
Terminal  : 


Jser  Information 

Name  :  hineline*  m. 
Lib  card  :  g»  h  ist 
Phone  : 
Address  :  0175-u 


«.«  }r~H 


m£i&'-         1 1 

:  A,  • 


■_ 
^^B 


